After fiddling with Catalina and LMS for a couple of days, I've finally got it starting, re-starting and running smoothly again. Here are some observations, for those still struggling after the OS upgrade:

Applications will dynamically appear in the Full Disk Access pane as LMS goes through its processes. You need to then unlock the pane as they appear, and then check those applications to elevate their permissions.

In Full Disk Access permissions, two LMS-related applications arise: "sh", and "perl". Once those two applications are given permissions under Full Disk Access, LMS starts and works normally again.

This is in addition to any prior terminal-based permissions commands that may have been required under Mojave. As a preemptive measure, it may be a good idea to re-run the terminal script application if Catalina has reset those permissions.

All plugins in my LMS implementation, with the exception of PlayWMA, are now starting and operating normally. PlayWMA currently relies on a 32-bit version of MPlayer, which is unsupported under Catalina.

Should I encounter any further glitches with Catalina, I'll supplement this thread.