I urge you to talk with your representative in your insurance company to learn about how these sort of claims proceed. I've been through this after accidents and know there is a procedure followed by all insurance companies. If you try to do this your way you will have difficulty. Let your representative talk with the other company.
I'm surprised that your representative (I'm not sure that's the correct title) didn't tell you not to talk with the other company. I've settled several claims, both when I was at fault and when I wasn't and my insurance company always told me to not talk with the other company. And I didn't. It's best to let those who know the procedure to handle it.
As to whether or not you should ask for more, depends on whether or not you're going to have difficulty in the future as a result of the accident. Sounds like you injured your back and that can result in back pain later in life if the pain is gone now.
I was rear ended in a car accident 40 years ago and still suffer neck pain off and on. I settled for $10,000 in addition to expenses and wish I'd ask for more. I've lost time from work and difficulty in general.
Your insurance company can steer you in the right direction.
I eventually hired an attorney on a contingency basis which meant he took a percentage of what I received. Eventually we settled out of court. I learned that insurance companies stall and frequently do not settle until a couple of years later. It was only after I hired an attorney that we closed.
You can consult with an attorney for no or very little fee before you decide whether or not you want to hire him.
Above all talk with your insurance company and do as they suggest! It is to their advantage to have the other company pay. They are on your side.
Do not expect to hear from the other company even tho you've sent them receipts. It will take time to settle. Do not settle until you are completely satisfied. The other company will put pressure on you to settle and not responding to your receipts is one way to do that.