Phals don't really mind being repotted while they're still in bloom. I've done it before numerous times. Just go easy on their roots and they won't care. In fact, I have a few that I rescued from the clearance shelf that I plan on repotting. My local Wal-Mart had just placed every Phal they had out in the full sunlight and marked them all down. I bought several of them and still felt bad because every single one they had was still in excellent condition and very healthy. But I knew they wouldn't last very long being blasted with full sunlight. Fortunately, I just happened to show up right after they moved them outside.

Just wanted to post an update photo of my little plant. The three new rosettes have gotten a bit bigger and are actually starting to look more like compact rosettes. They also each seem to be putting out some air roots. The mother plant's leaves are looking a little worse for wear :) I also noticed that the mother plant seems to be putting out two new leaves in between a few of the old ones. I know it will still be a while before I can figure out the ID (and I may never know!), but it is looking a bit like green goddess as Lisa suggested above. When I was searching green goddess the other night, an image of another plant came up whose little rosettes look similar to mine--though this may be just because they are so young. It was echeveria Van Breen (https://thesucculentproject.files.wordpress.com/2014/10/img_6541.jpg)

It had a partially ripped off price tag that had something like '....anthus' or something. But I searched for a long time and it kept showing non-tilly plants : ( Brachycaulos looks like it still has quite a bit of that fuzzy trichome surface, this one is actually really...bromeliad ish? so smooth/plasticky instead of fuzzy?

It looks a bit like a very etiolated Crassula elegans.....although the Osclaria deltoides is a possibility. Regardless of the ID, I'd seriously consider changing the potting media to something specifically formulated for cacti and succulents. The MG soil (especially the moisture control formula) is far too dense and water retentive to make these guys happy.