Hi fellow autistic 😉 (realise its a bit late but might help peiple who come across this later)
Once I had my DSA2 letters (nightmare trying to add certain things, and I had to add a new address to my letter becuase of times road closures which took 6 months because DSA hadn't reminded me that I needed to reapply so I thought it had been done etc. Also tricky as when i was trying to do some of it my assessment centre was in the process of closing down) and my taxi account set up, all I've had to do is log into the Addison Lee app, set it up with the code they gave me, and now when I travel it's set on my account and my mum's card is charged public transport (£1.64) and Addison Lee invoice the rest direct to DSA.
I personally would use Uber or something; that way you actually have digital proof of the receipts, but that's just becuase I would very easily lose paper receipts.
Actual mission, though because I'm on two different campuses throughout the week and my Addison Lee taxis only work for one campus and not the other or the additional address I have on my DSA letter, so I have to book Ubers/Wheelys (more Wheelys to pick me up; as they're much more reliable when prebooking) and claim the money back. It also took me the best part of a year to set up my taxi account and I've only been able to use it once a week this year, instead of 3 times a week.