Ouch, that can be complicated. I used to use SuiteCRM on Bitnami but gave up precisely because of the difficulty upgrading the core components like PHP, Web server, database… it ended up being much easier on plain Ubuntu. But then, I wasn’t using shared hosting.
You might try asking at the Bitnami forums, or asking your hosting service about it. At least you can go up to SuiteCRM 7.7 with your version of PHP.
A different route could be:
- set up another server with newer core components
- install your current version of SuiteCRM there
- migrate all the data and customizations from one to the other
- now upgrade the new server to the new SuiteCRM
The key point to remember is: don’t migrate and upgrade at the same time. If you’re migrating, stick to the same SuiteCRM version on both sides. “Server migration” and “server upgrade” are two separate moves, even if you’re doing one just because you need to do the other.