Lucky you having a 2 hour round trip as this year when I noticed goat in season I phoned stud goat owner, drove for 2 hours after work. Goat in with male 5 minutes 2 jumps,back in float , had cup of tea & chat with stud goat owner paid stud fee and received service certificate, then go the 2 hour journey home. Some people have to go further.
You could take her pal in the float for company and leave her in float whilst your other goat going to the male would be only with the male for a few minutes then paperwork head home.
Some people arrange to board their goats with the stud owner just before they are due in season and pick them up once served usually extra fees.Some stud owners don't always board milkers so may be an issue if both went to board.
AI is generally done at innovis centres at Edinburgh, Aberystwyth, Malvern some vet practices do it,
Some goat keepers have done a course for it cervical AI but can be as low as 20% success with frozen semen also transport of semen you want can sometimes be difficult. innovis are specialist vet team who do laprascopic AI which has a higher success rate some say about 60% with the semen already stored at their centres.