I don't understand, you can't contact the cab company until Tuesday? How do you book a cab then?
Over here, every cab trip is logged. I would only need to say "one of your cabs picked me up at X Street at 4.35pm Friday and dropped me off at ## street at 4.48pm." they have GPS tracking etc. Shouldn't be too hard.
I understand NYC is huge so maybe there'd be 20 cabs fitting that trip situation, just thinking out aloud.
That stinks, we need specific musician's insurance which covers our gear at gigs and in transit. There probably is such a policy but it's too much money for us poor musos lol.
I hope it works out well for you.