Insurance companies cooperate with each other settling property damage claims because they want to be cooperated with when the situation reverses. Your company is not in the game so that courtesy will not be extended to you. The company for the responsible party can make you get a lawyer, which will cost you money. Therefore, they will negotiate knowing that you may have that weakness. For example, they offer you $2500 for a 'contested' claim as a 'compromise.' You get a lawyer and they'll give you $4000, but the lawyer wants $2000.
If the numbers do not justify having legal help, consider immediately demonstrating that you will be a pain in the neck. What is the value of the total loss? Look at the maximum recovery limits to small claims cases in your jurisdiction. Maybe you can file immediately for that amount. The opposing company is going to have to deal with the suit like it is a big deal by hiring a lawyer and spending its money, and that might escalate the file to where someone will actually care about satisfying you.
It is amazing how many small cases become personal injury claims because the victim driver can't get satisfaction for property damage and then decides, maybe I will go see a doctor, it might not hurt that bad, but if I have to sue those jerks let's go firing both barrels.
Good luck.