I always recommend to use your AV system only for the karaoke music, not to sing through. It was never designed to be used this way. You could damage your equipment, especially your speakers. It's just not worth the risk considering how inexpensive a powered PA speaker costs.

A typical powered PA speaker would be something like this

https://www.amazon.com/Alto-Professional-TS408-Bluetooth-Loudspeaker/dp/B0BB15J4TZ

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3 or 4 wireless mic set up

I like this brand although I've never used this particular model

https://www.amazon.com/innopow-4-Channel-Microphone-Distance150-200Ft-Frequency/dp/B07W5PKR5F

song selection/control on iPad or laptop - I recommend using karafun run on a Ipad or laptop. Singers can browse the catalog or select songs using their phones.

video/lyrics display on TV - karafun has the ability to do this, especially using the HDMI output on a laptop.

equipment (mic receiver and mixer??) mounted to existing AV rack

Most mic receivers should easily be mounted inside a rack.

But the mixer is going to be a challenge. I can think of only one way to accomplish this.

A digital mixer with the ability to be controlled remotely like this https://www.amazon.com/Behringer-XR12-Tablet-controlled-Digital-Mixer/dp/B010NR6FS0

An alternative would be using a typical mixer mounted on a sliding rack shelf like this

https://www.amazon.com/Raising-Electronics-Sliding-Server-Mount-Adjustable/dp/B07JN8TG9Q