I don’t have a mac, and I don’t use iTunes anymore, so I can’t help too much there unfortunately…
Generally speaking though, iTunes will sometimes cause you problems that are beyond the scope of Picard/MusicBrainz - maybe someone with a mac has some other programs you could try (or not, no stress!)
You should definitely be able to select a few albums at once though? You can surely click and drag to select multiple folders, or click one, and then hold the shift or the command key and then select others.
Otherwise, drop everything into Picard, cluster it, and then use shift + click to select multiple files to run lookup/scan on.
My music collection is sizeable as well (and both of ours pale in comparison to some other digital hoarders), but unfortunately the answer still is that you have to work in batches, and not save everything at once. I know that’s not the answer you wanted, sorry!!
But it’s still much easier than doing everything manually 