Auto insurance companies are scum. Their primary objective is doing everything they can to pay the lowest amount on any claims. In fact, most have computers (ie Colossus) with brutal and compassionless algorithm to help them achieve this goal. Once an insurance company has accepted liability and its computer rendered a settlement amount, generally it'll be extremely tough to change it, even if you threatened to take it to court. It may change a little bit once a lawsuit is filed because it knows that you're already on the hook for about 1/3 of the settlement money in attorney's fees. It'll definitely change a few weeks before trial date which could be 2 to 5 years from the date of the accident. This is because it knows that you're on the hook for the 40% attorney's fees by going to trial and the insurance company finally has to assign councils to prepare and try your case in court.
It seems like your case involved only property damage so no personal attorneys should ever be needed. Let the insurance company handle it, pay the deductible and then take the other party to small claims on your own if you think it's worth the time and effort to right the wrong. Btw, you won't get a new door since the damage is minor. Even if it's a major damage, they would source a used door from their nationwide salvage-parts network before buying a brand new door. Typically, once the adjuster has written up an estimate, most bodyshops will work with that or they won't get paid for going over unless it's approved. You'll have to pay the difference on the unapproved amount and then fight it out with them.
Lastly, turn on your phone's video camera immediately after an auto accident and leave it running discreetly in your pocket. It may help you in the case of he said/she said event...