I've been vermicomposting in small apartments for year; and, in general, if you build it - and stock it with worms and compostables, and keep it warm, cool, & moist enough - they will compost. I have a balcony in my present apartment, where it's fairly easy for me to keep a worm bin in a 5-gallon bucket. In my previous apartment which had no outdoor space, I mostly used 3.5 gallon buckets, but did have a 1.5 gallon worm jar going for a while.

The smaller set-up was more sensitive to temperature and moisture level variations, and so I had to keep a closer eye on it; but it worked reasonably well. IMHE, there does seem to be a fall-off point starting somewhere around 2.5 gallons where the efficiency of composting deceases very rapidly with decreasing size. That is, while two five-gallon buckets compost not quite as fast and thoroughly as a 10-gallon bin, two 2-gallon buckets are a great deal slower than one 4-gallon bucket.

All my worm bins & jars have been plastic. I generally sift & restock every 1-3 weeks or so, and that seems to keep things adequately oxygenated.

TBH, I'm having a little bit of trouble visualizing your proposed set-up - could you post a a picture or diagram? Even just a quick-&-dirty sketch on the digital equivalent of a paper napkin would help a lot.