[Refplace]

You ask some tough questions so I tend to read more than respond. However though I have no relevant or current inside knowledge I'll take an uneducated guess here.
The CIAs mission is national security so he bulk of its efforts are focused where threats are perceived. I suspect the FBI has more assets focused towards the Caribbean, though both will be on the lookout for unusual expenditures, arms purchases, and anything that looks like money laundering.
Also an eye on governmental stability, especially for Cuba and any heavy oil producer. Sea and air traffic will be monitored by the Coast Guard and NSA but mostly looking for drug traffic. Still there going to notice any warships.

Given your background I would expect no serious competition, but they would be watching him. Once they figure out hes not funding a private army and just seems to be a nut job (since they dont believe in the supernatural) I expect it would mostly be hands off (he is a rich Texan and likely politically connected) but still watching for arms deals and major transactions.
That could limit his sponsorship at inconvenient times.

He should be able to keep hassles to a minimum with some political grease, high profile charity projects, public digs (after things have been cleaned up), etc.
Installing a few museums or such could keep locals from asking too many questions as well. If he is seen as good for the economy locals will likely help cover or even warn him if someone is sniffing around.