Even $400 for the Foredom is a "Yowie!" in my book. Unless this is
going to be a business.

I have a Makita electric die grinder ($70 ?) and a bunch of 1/4" carbide
bits that I got on eBay (10 for $10 ?). For cutting tile, I mounted
the grinder upside down under a table with the bit sticking through
(think table mounted router). This way I only have to hold the tile,
and I have the table to hold it on. Very stable. Holding a grinder in
one hand and the tile in the other is asking for trouble. I have a hook
up to my shop vac for dust. And I have a 1/4 - 1/8 adapter for Dremel bits.

This doesn't address your noise concerns, but I hope it helps.

Bob