Just following up on this. I did speak with VEEAM support and he was talking with the "Restore Team" for suggestions. Unfortunately, they did not have any hotfixes available for this issue and really didn't have any suggestions.
However, through a lot of trial and error, I think I have a pretty workable solution.
What I've done is, if I install the VEEAM Console on the SQL server that I'm recovering data to and set that server to the Staging server and select the EXPORT option, then it completely uses FLR and can do point-in-time restores from C:\VeeamFLR. My tests on this have been going very fast so far.
The only downside to this is that the VEEAM console has to be installed on all SQL servers, all restores will have to be done on from SQL server it's going to be restored to, and I'll have to make sure I have I have adequate free space for the Export restore if we don't want to overwrite anything.