Be careful with the RAC.
This from their T&Cs section 'Significant exclusions and limitations' on page 8:
"- If the Vehicle should break down whilst towing a
caravan or trailer We will recover Your Vehicle
together with the caravan or trailer. If Your trailer
or caravan breaks down then We can arrange for
the recovery of Your trailer or caravan but You will
be liable for the recovery costs."
As a friend of mine found out a few weeks ago. Ended up ringing a local garage. On his way home to Cornwall from Norfolk after collecting boat & trailer following a refurb of boat & trailer. Three tyres (one on each new suspension unit & the third as a spare) fitted to his trailer by the boat yard.
Two blowouts & having seen the dates on the tyres they are new, but manufactured prior to 1999 - he's currently having discussions with the boatyard about the cost of the recovery & a full set of tyres.
ETA for the AA members I'd want to have something in writing - an email for example - that their service covers the trailer if the trailer is the issue & the car is fine as their T&Cs are a little unclear.