If you try to use bluetooth for karaoke you'll have a noticeable delay between the lyrics and the music. You might decide it's good enough but you'll get a better experience if you connect the audio from the phone directly to the PA.
There are all-in-one powered speakers with mics. They might work for you but I don't recommend them because the mics aren't very good quality.
https://smile.amazon.com/Rockville-Rechargable-Bluetooth-RAM15BT-V2/dp/B07P2DW1VY?th=1
The other potential drawback to this system is the controls are on the back of the speaker. Depending on the setup it can be difficult to reach the controls.
Here is my post for a cheap home setup.
The best quality system in your price range would be this for the powered speaker
https://www.amazon.com/gp/product/B084KZCY54?th=1
and these mics
https://www.amazon.com/innopow-Microphone-150-200Ft-Continuous-WM-200-New/dp/B07F58YP2Z
This is the set I’ve been working with.
I bought them when I was short on cash and I needed better speakers and didn’t want spend the money I had on a mic.
Never had an issue, but I’m sure something would happen sometime and I’m always sure the Shure mics have much better sound.