Today, I've replaced server SSL certificates with new ones issues by GlobalSign. These should not suffer of same trust problems as CACert one used so far (especially after CACert root certificate being removed from Debian).
While doing this, I had to use SNI on server to be able to decide which SSL certificate it should use. This should work for any decent browser, but I guess your scripts might have problems, but I hope this will be rare. Anyway if you will face some issues because of this, please let me know.
Other than that I've also tweaked SSL setup to follow current best practice, what could also cause troubles to some ancient clients, but I hope these are non existing in this case :-). See Qualys SSL report for more details.
Anyway thanks to GlobalSign free SSL certificates for open source projects you can use hosted Weblate without any SSL warnings.
PS: Similar change (just without SNI) has happened last week on phpMyAdmin web servers as well.