I don't recommend using Bluetooth. It has connection issues and introduces a delay.
But if you're happy with it I would recommend new mics and a mixer with vocal effects.
Here is what I recommend
https://www.amazon.com/innopow-80-Channel-Microphone-Continuous-WM-333/dp/B0749L15PF
I use these mics and they are inexpensive, durable, and good audio quality
https://www.amazon.com/Yamaha-MG06X-6-Input-Compact-Effects/dp/B00I2J4V5G?th=1
I don't own this mixer but it's highly recommended by others.
You would plug the outputs of the Innopow into the MG06X and plug one output of the MG06X into the Rockville.
Your speakers are fine for the karaoke music but I don't recommend singing through them.
Most people are not audio experts, they just want to grab a mic and sing. If they don't understand concepts like stage gaining you could damage your AV speakers if you try to sing through them.
I recommend a mixer and a powered speaker to sing through.
Because the powered speakers I recommend they already have a simple mixer so you don't absolutely need a separate mixer. But I recommend one because they have vocals effects and it's more convenient having the controls close to you instead of reaching around the back of the speaker all the time.
Mixer
I don't own this mixer but it's highly recommended by others. I recommend it because you can use standard XLR cables to connect it to most powered speakers.
https://www.amazon.com/Yamaha-MG06X-6-Input-Compact-Effects/dp/B00I2J4V5G?th=1
Speaker
https://www.amazon.com/Mackie-Thump12A-1300W-Powered-Loudspeaker/dp/B073V7MKQZ
https://smile.amazon.com/Rockville-RPG10BT-V2-Bluetooth-Wireless/dp/B084KZCY54?th=1
Unless you have some other requirement for more channels the Yamaha 10XU seems overkill. The MG06X would be a more portable alternative.
https://www.amazon.com/Yamaha-MG06X-6-Input-Compact-Effects/dp/B00I2J4V5G
I have the same Innopow mics and they work well.
The Monster gives you the portability you require.
One other system I posted about you might want to look into.
https://www.reddit.com/r/karaoke/comments/smqble/my_2nd_recommendation_for_low_cost_home_karaoke/
1) not required but I highly recommend. I use these and they work well for karaoke https://smile.amazon.com/innopow-80-Channel-Microphone-Continuous-WM-333/dp/B0749L15PF
2) Bluetooth introduces a delay that makes it unusable for karaoke
3) OpenKJ is a great piece of software. I wouldn't invest in anything else unless there is an option you can't live without it doesn't have. However I recommend the Karafun subscription service and their software.
4) For a budget mixer I recommend these because they have vocal effects and XLR outputs
Yamaha MG06X
https://smile.amazon.com/Yamaha-MG06-6-Input-Compact-Stereo/dp/B00I2J4V5G
or Mackie ProFX6v3
https://smile.amazon.com/Mackie-ProFX-Mixer-Unpowered-6-channel/dp/B07XC3BHTL
For speakers I would recommend
Alto
https://smile.amazon.com/Alto-Professional-PA-System-TS310/dp/B07CKBWYV5
or Mackie
https://smile.amazon.com/Mackie-Thump12A-1300W-Powered-Loudspeaker/dp/B073V7MKQZ
Finally I would recommend using the Karafun professional subscription service and their KJ software instead of OpenKJ. It's free to download and they have some free songs you can try it out with before subscribing. Their catalog is updated regularly so you don't have to worry about buying the latest releases. You can use your own external hard drive of songs in addition to their songs but I don't use my external hard drive much these days.
Yes, you need a mixer. Something like this
https://www.amazon.com/Yamaha-MG06X-6-Input-Compact-Effects/dp/B00I2J4V5G
You would plug the audio output of the laptop and the mic into the mixer and the plug the output into the Rockville speakers.
If you want to go super cheap I have this one and it works well enough.
https://www.amazon.com/Pyle-Compatible-Professional-Interface-Mixing-PMXU48BT-5/dp/B07Y8M52HN
I would upgrade the mics to something like this.
https://www.amazon.com/innopow-80-Channel-Microphone-Continuous-WM-333/dp/B0749L15PF
Your mixer is fine. But if you wanted to upgrade I'd go with something like these
https://www.amazon.com/Mackie-ProFX-Mixer-Unpowered-6-channel/dp/B07XC3BHTL
https://www.amazon.com/Yamaha-MG06X-6-Input-Compact-Effects/dp/B00I2J4V5G
If you have good internet, you don't have to download the Videos, but it would make your setup more mobile and fool-proof. I don't see an easy way to do pitch control on videos. (others may have an idea...I don't)
You can connect the output of your computer/smart TV to a mixing board along with one/more microphones. The output of the mixer goes to your PA ( amp/speaker or powered speaker). If you don't have a mixer, Amazon has relatively inexpensive Behringer and Yahama units with effects that I've used and recommend.
https://www.amazon.com/Behringer-XENYX-QX602MP3-6-Channel-Player/dp/B01B7QKLWC
You don't need karaoke software as you aren't using karaoke files. Some karaoke software does have the means of playing videos but that just complicates things. If you are playing the files online, you can just use your browser. If you want to download them, use a shareware video downloader like 4k Video downloader ( https://www.4kdownload.com/products/product-videodownloader ) and a video player like VLC. I've used the free version of 4k Video downloader and it works fine without having to purchase the "pro" version. Let me know if that's not clear and I'll try to help.
OK
A typical system I recommend would be
mixer
https://www.amazon.com/Yamaha-MG06X-6-Input-Compact-Effects/dp/B00I2J4V5G
speaker(s) - you would need to buy speaker stand(s)
https://smile.amazon.com/Alto-Professional-TX310-Musicians-Ceremonies/dp/B097HZT1M9
mics
https://www.amazon.com/innopow-80-Channel-Microphone-Continuous-WM-333/dp/B0749L15PF
TV - I normally like to use 2 monitors, one for the singer and a bigger one for the crowd. I don't have a particular brand or size.
I use these to mount the TV / monitors
https://www.amazon.com/gp/product/B0057PQ8R2
laptop
Needs to have HDMI output. I change the audio setting to run the output in mono.
IMO you definitely need a powered speaker. As you found out the sound bar is not designed for singing. And the mics you have don't have the best audio quality.
Focusrite Scarlett 2i2 and Aspen Spirit mic are fine for recording but not the right equipment for live karaoke.
If you have a Shure wireless mic you need a Shure receiver.
Here is what I recommend
https://www.amazon.com/innopow-80-Channel-Microphone-Continuous-WM-333/dp/B0749L15PF
https://www.amazon.com/Yamaha-MG06X-6-Input-Compact-Effects/dp/B00I2J4V5G?th=1
https://www.amazon.com/Mackie-Thump12A-1300W-Powered-Loudspeaker/dp/B073V7MKQZ
You don't need two powered speakers if you run the karaoke music in mono.
That's an interesting question. Being able to spend more money opens up more options. Based on my hands-on experience and what I've read from others, here are my recommendations.
Mics - I've had these for two years and I still love them.
https://smile.amazon.com/innopow-80-Channel-Microphone-Continuous-WM-333/dp/B0749L15PF
Mixer - I don't have these but they have the features I recommend. I like using a separate mixer because you can locate the controls in a more convenient location.
https://smile.amazon.com/Yamaha-MG06X-6-Input-Compact-Effects/dp/B00I2J4V5G
https://smile.amazon.com/Mackie-ProFX-Mixer-Unpowered-6-channel/dp/B07XC3BHTL?th=1
Speakers
If you want an excellent small profile speaker and are willing to give up having a separate mixer then this speaker with mics is all you need. I've used this speaker and was blown away how good it sounded for karaoke
https://smile.amazon.com/Bose-Bluetooth-Speaker-System-Battery/dp/B078RT7VRZ?th=1
If you're looking for a single speaker, I don't have hands-on experience with it but this speaker has all the features I'd recommend and its's well reviewed.
https://smile.amazon.com/Mackie-Thump12A-1300W-Powered-Loudspeaker/dp/B073V7MKQZ
If you want to use two speakers in your setup I recommend Rockville. I have them and they work well. For home use I'd probably recommend the 10" because it's a little more manageable size but I like the 12" version as well.
https://smile.amazon.com/Rockville-RPG15BT-V2-Bluetooth-Additional/dp/B088ZH5CGH
Here is another style of Rockville that uses similar electronics.
https://smile.amazon.com/Rockville-BPA10-Professional-Powered-Bluetooth/dp/B01C564PZ2?th=1
My friend also recommended me this mixer