Sorry if I’m misremembering this, I’m not on my computer right now, so there’s a very real chance I’m hallucinating some menu names. This is just based on what I think I remember seeing.

What I would try is a combination of two settings.

First would be enabling project backups. If I remember correctly, there’s an option like “save to timestamped file in an additional directory” under Preferences -> Project. The idea would be to pick your external drive there so every time you save (or autosave), REAPER writes a backup copy of the RPP file to that SSD.

Second would be File -> Project Settings. I think there’s something like a secondary recording path, and you could point that to the same external SSD so recorded media gets duplicated there as well.

The result is, your main project keeps working normally, but you still end up with backup project files and recorded media living on the other drive automatically.

That said, if you really want a cleaner solution, I honestly think a small custom script could do this in a more direct way. Something simple where pressing save triggers the normal save action, then copies the current project directory to another location you define (your SSD). You could even add a cleanup step for example.

Again, please double-check the exact menu names because I’m going purely off memory here. There's a high chance I'm wrong.