Currently Southwest Airlines has flights from LAX connecting through HOU to BZE for as low as $172 leaving and $240 returning depending on the dates. This includes two FREE checked bags per person.
The new Southwest Houston terminal is a breeze to navigate between the domestic arrivals and the 5 international gates, all within security going and an expedited path/baggage recheck on returning. I agree the MIA AA terminal can be a headache and a long long way between gates.
For just one month rental, I suggest using AirBNB or VRBO websites to find good rental prices/places for your high season visit. I have always been able to find good places for 2-6 week rentals anywhere i have been. For shorter duration rentals I prefer www.Booking.com.