That must be gut-wrenching, I feel it for you. I've never held any faith in any of the recovery services. Unless it's during normal working hours (i.e. mon-fri 9-5) on a main trunk route in normal weather there's little hope of them getting to you usually within an hour. I've never needed to use them when I've been in car (the only time I've needed car recovery I didn't have any sort of cover). Of twice needing to use recovery services included in motorbike insurance (in over 30 years) they've proven themselves absolutely useless. Last time stalled out within 6 miles of home on a Sunday afternoon (sunny) - They explain that they'll need to get a recovery contractor from Brum (to Northampton) and It'll be 2 hours (I could walk home in that time but not pushing a 750 shaft drive bike). About 40 minutes later a call to my mobile just get's missed as I can't hear it over the traffic noise - number witheld - put it down to a spam call and keep waiting. When time is up phone them to ask "where's the recovery?" to which "Oh, we tried calling to let you know it'd be another 4 hours as the original contractor couldn't attend..." . Yes, the spam call, it was them trying to call ONCE and once only no text to back it up no "sorry sir, we should have considered that you might be in poor reception area, not hear the phone ringing etc, or realised that we could've sent you a text" and no, for some reason they didn't think that their psss-poor effort at recovery taking over 6 hours just because it was Sunday was in any way absolutely useless.
Sorry, decided to rant as well cause they're all pretty pathetic these services - Anyway back to your issue - I'd suggest you make a huge song and dance about it and escalate it to anyone who might listen - consumer press etc. Car ruined through one small bump because AA is pathetic etc...
Here's a link to rip off britain 'submit your story' here
https://www.bbc.co.uk/programmes/b00wck32/contact
And the same for Watchdog
https://www.bbc.co.uk/programmes/b006mg74/contact
And just in case you're thinking 'why would they bother with my story?' remember that if you don't submit it they can't bother with it anyway.
So what's to lose? (and as a saying in our house goes 'Shy bairns get nowt'). Embarrass the useless clods of earth that are the AA Get photo's if you can as well TV shows love pictures...