My main concern is that I won't be given the chance to defend myself by the DMCA institution.
In practice, though, the one time I sent a DMCA notice to a hosting company, they tried to mediate between me and the site owner rather than immediately taking anything down.
Note that as far as I know, non-US companies will usually comply with DMCA requests.
No they won't if operating outside US jurisdiction. As a UK hoster we won't and don't recognise DMCA requests. We'll only remove content upon a receipt of a properly filed request/complaint by English lawyers (I work for a company registered in England).
Assuming that the theme marketplace is in the United States and that your competitor does file a DMCA, you have the opportunity to file a 'counter-notice' with the service provider (theme marketplace) asserting your work is original and that you have the right to distribute it. The service provider is supposed to restore your original work in 14 days unless whoever filed the original DMCA notice brings a lawsuit in court. Whoever files a DMCA notice may be liable for damages incurred by the takedown.
- Ask them what steps are necessary to restore in case it does happen.
- Ask them if there's anything they would like at this time to prevent a long takedown / restoration process.
If everything is as you claim, it's possible this is a common issue with the complainant and the final straw to force them to leave the service due to a pattern of harassment.