You can often find private individuals who rent their cars for long term periods, at a lesser price.
It's a matter of shopping around.
You can start by asking the "in town" car rental agencies if they know of any.
PS: Another alternative is buying a cheapo car and reselling it near the end of your stay.
I saw a tiny subcompact, 1998 (I don't know what brand it was) at Espaillat in Moca, last week. The outside looked great, other than it being so small that it could fit into the trunk of a Cadillac, and it was selling for 110,000 RD, which is equivalent to approx. $3,500 US.
I'm sure you could drive it for 3 months and resell it for less than $1,000 loss, if you lose anything at all.
There's always a market for a super cheap car.
They sell faster than the more expensive cars.