This sounds a bit dodgy to me and due to the persistence of the calls and messages I can’t help but wonder if it’s the other party hassling you making out they are the garage / insurer.
Whilst it is quite normal, and not unreasonable, for a driver to offer to settle damage that they have caused in order to avoid an insurance claim (which affects no claims etc), it sounds very odd for an insurer to also offer £200 to not use insurers!
Even if it’s legitimate, there is always a risk when accepting a direct settlement that there might be additional damage that you cannot see. The other party may not point this out and / or may not repair it / claim it was there before etc. Also, you may not be happy with the quality of the repair (I have rejected two repairs due to poor quality). This is easier to do when your insurers are behind you.
Unless it’s minor cosmetic damage (when I’d happily settle with the driver) I’d involve my insurers to cover myself. Unfortunately, whilst it’s a no fault claim, so won’t affect your no claims and won’t result in you paying an excess, it may still result in a slight hike in your renewal premium as insurers will weight anything and everything as that’s the way they are 😡