Yeah. The danger here is you pay out and then they suddenly get whiplash once they realise you’re going to go “oh, ok then.” Then you’ll be out of pocket to the tune of this repair plus your excess. Going through the insurance gives you, well, insurance against that sort of thing.
As for what you can ‘request’, you can request anything you like, you’re doing it unofficially between yourselves. They, equally, can tell you to get stuffed. An insurance claim would typically request at least two quotes so I don’t think it’d be an unreasonable request. If they say no, your counter is simple, “we’ll go through the insurance then.”
TBH, if they’re agreeing to this, then either they’re amenable sorts who don’t want to cause you unnecessary strife, or they don’t care and just want as much money as they can.