Two thumbs up for the OP taking responsibility while asking realistic questions. I'l buy you a coffee anytime.
Unless the other posters do circle checks everytime they enter their cage they are equally likely to back into something at some time.
There is no legal onus on a rider to park visibly any more than there is a legal onus for me to check for cross traffic when I have a green light. I'm alive today because I did.
Re settle / or insure:
I had a love tap from a taxi ($500.00) and they cut me a cheque with a disclaimer saying that by accepting the cheque there would be no further action on my part. I took the money and lived with the scratch on my cage.
My wife scraped a co-workers bumper paint at work and left a note. The co-worker got several very reasonable quotes and we paid cash as there were good vibes. We fixed our own damage.
I did a similar thing on a different occasion when I had a good feeling about the guy.
If I do $2000 damage to someones property, give them the money and they spend it on beer it's really none of my business. I did my part and they chose to live with damaged goods and a beer belly. I would want paperwork to keep them from coming back. If the damage repair estimate was reasonable and the guy fixes his own bike for free because he has a stash of parts or steals them it has nothing to do with me. I enabled him to restore his property. Sorry if he steals yor bike for the parts but I can't baby sit.
The concensus seems to be that the $2K is reasonable to fix the bike up. Whether there was prexisting damage is a crap shoot. It would cost more to research than could be justified. Microscopic examination by a forensics lab etc.
OP, what is your gut instinct? If you have doubts then you will sleep easier if you let the insurance company (with their legion of lawyers) do what you pay them to do. Hopefully as your policy matures the rate increases will mellow out.
Cheers and good fortune with your decision.