Involved in Bike Accident; Advice?

Hi, wanted to get the denizens of reddit to opine. I (and my kid) were involved in a bicycle accident, in Brooklyn, NYC. We were biking in a bike lane, got doored by a guy blindly opening his door which gave us no time to stop nor swerve. Both of us were thrown off the bike. 911 was called and police/ambulance responded. My young kid was complaining of abdomen and head pain so we were both taken to the ER, but thankfully nothing serious – no broken bones or anything except mostly severe bruising on my part (kid ended up being fine).

Apart from the pain (and thus not being able to work as much; as well as general inconvenience), this accident (which was clearly not our fault) will cost us ~$1-1.5k between medical bills and the bike repair. I'd like the other guy to cough up for this – question is what is the best way to go about doing this?

For reference, I have the police report (he stayed on scene with the police while we were taken to hospital). From that, I have just his license plate, name, address, DOB, and in the insurance section it has something called "vehicle number (it's an actual number, not a license plate), expiration date and VIN". What's the best way to go about doing things properly? My options as I see it are:

Send him a snail mail letter (I dont have his phone number) asking him to contact me to hopefully settle it amicably such that it doesn't hurt his insurance perhaps?

File a claim with his insurance? – although I don't know what insurance carrier he is with – I just have the vehicle number/date/VIN. Is there a DMV database or something where can look it up?

Get a lawyer involved – seems a bit extreme for what is not a huge $ amount.

FWIW I have Velosurance bike insurance which supposedly covers bike repair for accidents.

Would love to get recommendations on how to proceed.