I would suggest the OP buy a second small, used and somewhat inexpensive car for the second residence since this isn’t a one time cost…it’s every year and re-occurring. At the end of the day, that will likely be less costly than renting for four months every year.
I bought a relatively inexpensive car a few years back. I sold it last year for very little, as it need work. It was not a good deal as I was able to use it one year only—just 3 months and then sporadically. Cheap old cars need maintenance and repairs and are not reliable. Maybe, you get lucky and you make out. Maybe not. I was not lucky.
If I can get a rental for under $400, it would be ideal. Not gonna happen. Will just forage for rides. I can get buy on $10 day for jitneys and getting rides here and there. Inconvenient, but fine. Maybe a weekend car rental a month. I know people there, old times neighbors , cousins and have friends. $400-500 allocated per month will get me around though not with the freedom and convenience of renting my own car. But the numbers are not going to work out
If you could rent cars for $400/month, lots of folks would not buy cars 
Any chance you could find somebody who wanted to drive your car one-way? Then you could use your own car, without the chore of long drive at beginning and end of the gig.
Another idea, for a few weeks at a time … find a family (or retired couple) who is taking a long trip and does not want to pay for airport parking.
As mentioned up thread, my sister is in an expensive car rental for a few weeks while her regular car gets fixed. She has had some trouble with the rental car, and cannot figure out how to lock it as the battery in the fob has gone dead. In addition, the oil change light came on. The rental company has told her to buy the battery herself for the fob, and take it to change the oil herself. They would reimburse for costs. My sister works 40 hours per week plus, and cares for my mom after work. She is a bit technically challenged at times, and I am livid that this rental company is charging her a high fee and expecting her to take care of details of maintenance in her non existent spare time.
Opinions on these expectations? I have told her to take it up with her CC company as the local rental outfit is not reasonable.
Posted, as these are considerations in a longer term rental.
Are there any branches of the rental car company convenient to her workplace or home?
When we had problems with our 1 week car rental (slow leak in the tire), we called the tollfree number and were told to buy a new tire or fix it on our own dime.
We found those options unacceptable and went to the nearest Branch location of that car company. The branch apologized that they just gave their last car to another renter who was having problems with their car but we could take it to Big O across the street and have them fix it and put it on the rental car company’s account.
We decided that was likely the best of our options, so we did that. It did take several hours waiting until they got around to servicing our car (very busy with other appointments) and repairing the tire but we were NOT charged extra, whichvwiuld have added insult to injury and the car worked fine for the rest of our rental period.
It seems like cars are not maintained that well any more by rental companies and they seem to expect us who are renting to get the maintenance done while we are renting! They also don’t seem to have spare cars to replace those that should be serviced.
Yes, you kind of have an expectation that rental companies would send someone out to do minor stuff like change tires and batteries. But in reality either you fix it yourself and get reimbursed, or go through the hassle of driving back to the rental car center and swapping the car.
For a week’s rental I would just ignore the oil change light. A few days won’t matter and you’ve reported it the company so ethically I’d consider my responsibility to end there. For a low-oil light (different from an oil change light), I’d probably check the oil level and add a quart of oil if it was really low.
Changing the battery in a key fob is easy and just needs a small screwdriver. You’d probably find everything necessary in a large drugstore/Target/Walmart - grab an eyeglass repair kit (for the screwdriver) or a cheap screwdriver, take the fob apart on the spot, grab a replacement battery and insert on the spot. Then take the wrapping/tags to pay, and you’re on your way in about 15 minutes and $15. And you could probably grab a quart of oil there too.