I'm not 100% sure about the audio output on the TV. Are you planning on using the 3.5mm headphone jack or something else?

The MX400 mixer is small but not suitable for your situation. If you want something small this is something I own and it works well.

https://www.amazon.com/Muslady-4-Channel-Connection-Recording-Broadcast/dp/B07QGWG3H4/ref=sr_1_5?dchild=1&keywords=Muslady+SL-4&qid=1608209742&sr=8-5

Although it's possible to use a soundbar to sing through I don't recommend it.

The cabling I believe would be

TV 3.5mm stereo to dual monomixer ch3 & ch4>> headphone jackStereo 6.5mm to Stereo 3.5mmsoundbar