I know Logos has a copy of my library. Is there some way for me to access that copy so I can verify that we are in sync between my library and the one that they maintain?
Mark,
Assuming L3, you could run a license report (Tools → Library management → Account Summary)
Assuming L4 since your license file is synced on every program start - as long as internet is set to yes- it is no longer necessary, or possible.
Now then, if you install L3 as well as L4 you can Synchronize your licenses in L3 and then run the report to cross check your licenses.