Hello,
Does this happen on Windows? (It's OK if you don't know.)
If it does happen on Windows, it would probably make the most sense to try to contact the publisher ( contact@blackobelisksoftware.com ?), or to find an alternative workflow that involves running this program outside of Dropbox.
If it doesn't happen on Windows (or if you don't know), you can wait for a CrossOver update that may solve the problem. You can also still ask the publisher to change their .tmp files so that they also start with ~$ (a tilde and dollar sign) or .~ (a period and tilde). Files that do so are automatically ignored by Dropbox.
In short, the best solution is likely to find a workflow that allows the .tmp files to either be ignored or generated outside of the Dropbox folder, especially if the .tmp files are generated while on Windows.