its know as capturing the third party (direct).
Your neighbours Insurer is able to control costs by dealing with you directly, rather than waiting for a bill, for which they have had no consultation, to role up from your Insurer.
If you claim off your own motor Insurer your no claims discount (if unprotected) will be impacted, until the point of the third parties cheque arriving back with your Insurer. So if your renewal were on the horizon say within the next 3-6 months you could find your premium is higher as a result of this claim, if at the time of renewal your Insurer hasnt had the third parties money in by this time. Renewal is usually sent out 28days in advance as well.
Dealing with the TP insurer direct should be fine. Check to see what parts they will use to fix your car, pattern parts or original, the industry is awash with using non OE parts to repair cars – essentially sub standard parts which dont fit as well. Its reasonable to demand that your car be fitted with genuine parts.
Secondly, the hire car they provide is likely to be pants, think Vauxhall Corsa, if you went to a no win no fee solicitor you’d get something more akin to the car you normally drive.
This is a point of discussion, you have a duty to mitigate your loss (keep costs down), but if you are a family man, have a need for a bigger car (transport bike, etc) then ask for a car that is suitable, if they decline, say you will go to a no win no fee solicitor and I suspect they will re-think their answer.
Hope this helps, this should be easy as the neighbour had admitted liability so its only costs to agree.