As I read it, there's no restriction on a cleric joining the TigerEyes.

Is it correct that a holy-man can get into the TEs, despite there being potential conflicts with his religious objectives and the objectives ot the company?

I think it's something that could be discovered in character, though this may seem like a cop out - it's the same as with anything else. If you can provide a rationale ICly, then it's not going to get blocked for OOC reasons. (clerics of Bane becoming councillors slyly, strong wizards with martial weapons feats becoming watch members )

The Captain is his own entity, and i can't provide his views here, of course :)

One of our first Tigereye PCs was a cleric!