Not that I'm aware of, but I only use TL to archive my listings, and resurrect the occasional long expired one from those archives so not an expert in TL's nuances.
TL has no command line paramet options I can see, so it isn't possible to invoke it from a batch file or a scheduled task and get it to do anything that way except just sit there.
It is possible to do it using an Autohotkey (AHK) script as a scheduled task.
[An AHK script can be run by Autohotkey (interpreted mode), or the AHK script can be compiled to a standalone exe to get around having to install Autohotkey, but the compiled exe can't be edited to tweak timings or other things (original script has to be edited and recompiled)]
AHK script (or compiled exe) would be set up as as the monthly scheduled task. Script would start TL, send appropriate keyboard and mouse actions to TL to walk through the backup process, and shut down TL after.
Same type of thing could probably be done via Powershell (not fluent in Powershell), probably VBS, or maybe via a batch file with appropriate external utilities (PSList, PSKill, a sendkeys program, etc).
I have an AHK script in progress, but it has some reliability problems. TL is a pig, and doesn't start consistently. Sometimes takes 30 seconds to start, sometimes 3 minutes, sometimes doesn't start. Then about 10% of the time when it starts it throws up the "what do you want to do dialog" (open a new user, use an existing one, etc). Even after the window starts to load, it takes a while for it to fully load to the point where starting the backup can begin. Takes a lot of "if then" and "wait for window" logic to deal with it.
Since this is a scheduled backup approach (wee hours of the morning?), it might be better to to stop fighting the load times, try to start TL maybe 3 times, wait 5 minutes for it to load and stabilize, and then start the backup (have to look at that maybe tomorrow when time permits)
Also, it was unclear from your OP, but it sounds like you may have multiple profiles working in TL and want to back all of them up on a scheduled basis? That would complicate things, but doable.
Maybe 50 lines of code in the script, and about 200kb compiled as a standalone.
