It is safer to get it done at the vets, yes they put them to sleep under gas but that way they can also get the back very quickly too.
Mine travelled in a dog crate in the back of the car, and it only takes 20 minutes to get done! Be prepared for the look, and in particular for the smell afterwards, I couldn't cope with the smell for the first one, but was ok when the two girls went in a few days later.
Also vet put the ring on the wee male, as he was under it was much easier. I normally do all my lambs myself, so no issue in doing it, but vet asked and didn't even charge for it...
Also keep checking him a few weeks afterwards for any re-growth, my two BT girls have had two disbuddings already, and one looks set to go in for No 3.... not very happy about it, but nothing I can do (and the vet is experienced, he has done loads of goat disbuddings in the 1980s....). If you plan to keep the boy as a pet in particular, horn growth after disbudding can be very irregular.