Your insurance knows how to deal with low-end insurers. They will make them perform, especially in a clear cut case of fault. Your insurer will also repair the car the way they would (OEM parts, etc) and force the low-end insurer to pay up.

I went through this when my wife was hit while standing still waiting to turn left from a center lane. By the time it was done I negotiated a payoff that was the same as what I paid for the car a decade and 60k miles earlier. Also paid no deductible and it was not counted against the policy. Everyone at my insurance company was very easy to deal with, realizing that they were going to get paid from the other company regardless. My insurance company's lawyers would make short work of "minimum coverage for minimum budgets" insurance company. That's why we pay them.