Sorry, I find it almost impossible to believe you could rent a car in Playa for $466 per month. Don't believe that all needed insurance is included at that price. It is not difficult to make reservations for even cheaper than that, but somehow it just doesn't work out. Normal cheap car rental rates for "beater cars" from Less Pay or Isis will run about $200 per week. It's likely you could get some discount for a monthly rental rate and more so for an extended period. Will the four month period be during "high season"?
You could probably bring a car to the island, but it is unlikely you would really want to. Renting a beater car or even a non-beater, Whether it be a flat tire, a dead battery, expect something to happen. No mainland company will come to your rescue or assist in resolving any problems. Occasionally someone will make a post of a super cheap rental, most often booked online ($75 a week, $9 dollars a day!) from someone who mistakenly believes that they have insurance coverage through their credit cards or other means as they might inside the United States. They are risking a major, life changing event. If you find yourself at the police station after an accident, you want to have a local car rental agency representative or insurance agent present.
Just curious, where will you be staying? You think you need to have a car during your entire stay? My opinion would be to see what rate either Less Pay or Isis might offer. Fifteen years ago, I did have a deal with Less Pay by the month including a factura so it was a deductible business expense. It wasn't close to your stated "American" price. I doubt John could match my price today. My guess you will get the best price/value from Less Pay or Isis. It is what it is and this will be the price of a ticket to do as you wish. Be very careful with any prices that fall into the too good to be true category.