I've searched the net extensively without finding a definitive answer about whether swapping a 722.323 transmission for the later 722.350 version is
plug-n-play.

I just bought a used 722.350 MB factory exchange unit w/ 50K miles on it, from a member here, to drop into my '87 560SEC.
In all my online research, I never read that this swap was
not doable - other than the Hollander Exchange manual apparently does not list as exchangeable, due perhaps only to different 1st/2nd gear ratios in the 722.350, which I did find mentioned online.
By the same token, I have not been able to find
any reference to someone whose successfully made this 'upgrade'... and I had to ask:
How come ??