RØDE - Stereo VideoMic Pro

I used to use a Rode Stereo VideoMic Pro years ago and it worked very well for most situations. But, I have a Zoom XYH-6 capsule installed on my Zoom H5, it is much better than the original H5.

As others have said, get a decent external mic. I'd also suggest getting an external recorder that records 32 bit float files and then combine the video and audio in post. Something like a Zoom H1 essential. If you want to record good stereo then you can go down the rabbit hole as much as you like. Rode stereo videomic or the SENNHEISER MKE440 etc. Whatever mic you use however if you're not using a field recorder and are connecting the mic to the camera, use the mic on full output and turn down the gain in the camera. The preamps in cameras are usually pretty poor and the less work they do, the better.

A Rode VideoMic Pro or Rode VideoMic NTG would both be good candidates for the use you've described. The former should definitely fall within your budget; depending on where you are the later might be a little over.

Rode video mic. But it’s consumer grade. A big step up from phone mic for close (2-4ft). Not “top quality” because that implies you’re using a proper preamp

In my experience, awful for anything with extremes (really quiet or really loud) and even “normal” conversation/dialog doesn’t sound GREAT. I would definitely recommend an external mic. For karaoke specifically, the Rode videomic should be OKAY, without diving off into more expensive and cumbersome equipment.