You say your web designer says there are a great deal of problems, but what are those problems?
PrestaShop uses a combination of php and MySql, which many content management systems use. I can't see any particular reason why that should cause a "great deal of problems".
I note that PrestaShop says on their site there are issues with certain versions of php and Mysql.
Perhaps that's where the problem lays. That said, I use MySql version 5.0.51a and php version 5.2.6-1 on my Debian server without issues, albeit with WordPress.
I appreciate I'm not giving you a direct answer to your question, as I don't use PrestaShop and quite honestly, before your post, had never heard of it. But without knowing what your problems are likely to be, it's difficult to make any reasonable stab at the likely outcome.
If it's simply an incompatibility of the versions of php, MySql and PrestaShop, you may have no choice, but to use a different host that is compatible with PrestaShop.