No, it is on a shared hosting, but a reliable one. Probably there are some problems with the modules, but with the theme as well. And it seems much too much work to try to figure that out.
Since the shop is currently running stable PS 1.6 on PHP 5.6, my suggestion to the site owner was to keep it like that for the time being and to start working on a move to Woo. So it seems we will completely skip the Thirtybees experience. Only in case that the hosting company will completely stop support for php 5.6 before we're ready, we will then, as a temporary measure, go to PHP 7.4 and migrate to TB 1.4
Thank you all for the help!