If you use a shared system then you are at the mercy of both (a) the expertise of whoever configures and maintains it, and (b) the behaviour of others on that shared system. Chances are high that it's poorly configured and maintained and that other on the same host are spammers, or their systems are infected with spam-sending bots - so your IP gets on all the blacklists.
If your client really cares about getting mail through then they will be happy to pay for a service, or for you to work though a good setup of postfix on a VPS. If they're not willing to pay for this then don't stress about it.
Comments
If you use a shared system then you are at the mercy of both (a) the expertise of whoever configures and maintains it, and (b) the behaviour of others on that shared system. Chances are high that it's poorly configured and maintained and that other on the same host are spammers, or their systems are infected with spam-sending bots - so your IP gets on all the blacklists.
If your client really cares about getting mail through then they will be happy to pay for a service, or for you to work though a good setup of postfix on a VPS. If they're not willing to pay for this then don't stress about it.