You can't. Not directly. N-regs can only be owned by people or corporations that are legal residents of the US. Or something like that.
The solution to this is to establish a US trust. This then legally owns the aircraft, but as part of the trusts regulations you are the beneficiary.
Amongst others on here, IO540 operates an aircraft under these conditions. If he doesn't step in all by himself, you might want to send him a PM. Or simply:
Let me google that for you
The other things, obviously, is that you need FAA licenses, including current BFRs and medicals. But you seem to be well informed on that front.
But we really want to avoid importing it to the DR and putting it on the Dominican register.
Bear in mind that "importing" and "putting it on the DR register" are two totally different things. Depending on the exact regulations, even if you leave the aircraft on the N-register, you might still be liable for import duty/VAT in the DR. In order to avoid import duty/VAT you typically have to take the aircraft outside the country for X months during a year.