Samuel Sieb:
I think, by default, Thunderbird puts temp files in /tmp. Would a program running in a snap even be able to access that?
Richard England:
Thunderbird lets you define where the downloads go.
See Edit > Settings > General and scroll down looking for "Save files to". Check it and select/browse to the location you want to save to.
How browsers (and other browser-like applications) deal with temporary files (click and open directly) is usually different from how they deal with download and save.