Would one of these work?

https://www.amazon.com/BONAOK-Bluetooth-Microphone-Christmas-Smartphone/dp/B0773M39P6/

https://www.amazon.com/Microphone-Wireless-Bluetooth-Portable-Handheld/dp/B07TS5RYML

They're both under $100, can record, and have a micro SD card slot.

Based on the equipment you've provided you have no analog audio connection between the TV and the Rockville. The karaoke music will play only through the TV speakers which is perfectly acceptable but it won't get super loud. For a small room this might be adequate.

If you want to play the music through the Rockville you'll need an analog audio output on the TV. You didn't supply the model of TV so I can't give you a specific recommendation for cables or additional equipment. The audio output on the TV would be plugged into the Yamaha mixer or directly into the Rockville. You don't need a 2nd mic input on the Rockville. The Rockville has Aux and Line Inputs you can use.

I prefer wireless mics but the wired mics you've chosen will work just fine.

Based on the equipment you've listed here is a possible setup for a small living room.

You play the music through the speakers on your TV

You connect the mics to the mixer with 2 XLR cables.

You connect the mixer to the speaker with a single XLR cable.

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If the room is small and you don't need the volume very loud you also might consider getting some karaoke microphones. These aren't very loud but might be good enough for a small room. With these you don't need any other equipment.

https://www.amazon.com/Microphone-Wireless-Bluetooth-Portable-Handheld/dp/B07TS5RYML