Is it possible to have MySQL and PHP run with one provider and the "database driven website" with the other? Say with the help of only SSI / CGI? Print

Yes. Let your CGI/SSI include from the urls on your other site. On your other site you return incomplete html, ie. never return, and then you just provide the functions you need as different url on your other site.

More advanced solutions would be using http://www.xml-rpc.com or http://www.soaprpc.com.