Auto electrician or mechanic?
A mechanic covers the mechanical side and plenty of common electrical work. An auto electrician is who you want for wiring, charging faults and anything intermittent, which is exactly what gets referred on.
Why does my battery keep going flat?
Commonly the alternator, the battery itself, or something drawing current while parked. Those are three different repairs, which is why diagnosis is charged as its own step.
Is diagnosis charged separately?
Usually, and it should be. Ask whether the diagnostic fee comes off the bill if you go ahead, since many shops do that.
Can they wire a towbar?
Yes, and it is one of the most common jobs. Modern vehicles often need a bypass module so the car recognises the trailer, so say the make, model and year in your request.