Can a flat bed towing service tow a 13 ft camper

Flat bed or roll back?
Roll backs have short bed areas and the trailer must fit with the tongue to rear wheels on the bed and can hang over rear. Some towing services will not do this.
A flat bed is a tractor trailer type set up and can haul nearly every type of rig with no problem.
AAA normally does not have flat bed setups they have roll backs. They contract that to heavy towing companies. These are normally for commercial "recoveries".

Can they tow it... Depends on the drivers.. I know drivers who actually know what they are doing.. I know drivers who .. Well let's just say they have a few problems...

I was (For a short time after I left the State) A tow truck dispatcher. There was more than one time I had to tell a tow driver how to hook a car (or in one case my motor home) to get the job done... Now I know I am good. but the only time I ever drove a tow truck, It was one way (Back to the yard with a disabled driver riding shotgun) and empty. So I should not be able to tell a driver how to do the job (I do admit to being trained in science and electronics, including math and physics, And being able to see in 3 dimensions, which is why I can see how to do it when they can not).

Flat beds usually have a ball hitch that extends so they can carry a vehicle and a trailer.

This is if the trailer is safe to tow in the regular manner.

Flat beds have carried disabled 35-40 ft motorhomes. Surely can transport that trailer. I would look into Coach Net roadside assist rather than AAA. I think you get more bang for your buck.