It might be a theme specific issue or might be caused by some plugin. you have to check one by one.
01. Try deactivating all plugins then try saving theme options.
02. Try to use a default theme and try to save options, what happens?
03. Try clearing your browser cache and do the same again.
As you said this is from mythemeshop, you can also ask in their forum.
I have another suggestion for you, to check if your .htaccess is written properly or anything is blocked by .htaccess
Lastly, you can contact namecheap support if they have any mod_security triggered for your domain. I guess they can solve your problem as you said the same theme is working just fine on siteground.
Sometime hosting block some functions for security and shared users are not able to whitelist those.
In some cheap hosting the memory limit can also be a problem sometimes.
I think it's always trial and error to find what is causing the problem.