At 6 I would have thought a 16″ wheel bike is going to almost be outgrown although if they aren’t really confident riders a 20 might be a bit hard to handle when not riding.
Our youngest turned 6 a couple of weeks ago and is now riding a 20″ specialized Hotrock, her legs could do with being half an inch longer (or the crossbar half an inch lower) for standover but we went for a 5 mile ride earlier in the week and she kept up fine.
We’ve got a purple/lilac 16″ hotrock in fantastic condition which is now surplus to requirements. It’s going on the classifieds later in the week once I’ve taken some pics. It really is like new other than slightly grubby grips and would be under your budget if you changed your mind on new :)
edit: lots of other posts while I was typing this one, I see why you don’t want 2nd hand with twins :) I’d definitely look at 20″ wheels if you’ve got to buy two, step through style frames are probably better if the sizing is marginal as they will be able to step forwards off the saddle and comfortably stand flat footed.