It's a shame that you don't like your vet. Most good ones, you can just call them and tell them you need tranquilizers and they will just give them to you without having to bring them in. Our vet is like that.
Maybe you can try Spirit Essences herbal sprays. I know they make one that is for calming.
I really do hope that you find a better vet. It is so important for us to feel comfortable with and be able to trust the doctors-for us humans, too.
One last thought -- make sure each cats' carrier is large enough for the cat to stand up and stretch and turn around. My dad used to complain that his cat hated his carrier. When I asked about it, turned out that Dad had been using an under-airline-seat carrier they got once when they had to fly their small dog from one place to another. No wonder the cat wouldn't come out for a day after having been driven for hours in a carrier that small.
Also, I assume you are having the kitties travel in the pressurized baggage hold, since there are three of them and most airlines only allow one cabin cat. If that is the case, I don't think you have to take the cat out of the carrier when you check them in. (Each cat is in his or her own carrier, right? You're not thinking of putting two in one carrier or anything like that? I don't think the airlines allow that, frankly.) Anyway, if you don't have to take the cat out, that's all to the good if you have a cat who will freak. Last time I traveled with my dog by plane, the airline took charge of him in his carrier before I went through security. I assume they X-rayed him with all the baggage going by. I certainly didn't have to take him out of the carrier. Basically, a baggage guy was there to take him down the freight elevator when I checked my bags.
Be sure the cats are familiar with their carriers well in advance and find them comfortable places to hang out. If they think the carriers are a safety zone, that will help. I have flown with my cats lots of time and never medicated them, but these were also cats who were used to being in a carrier, and used to being in a carrier in the car. Do you have any time between now and the trip to pack them up and take them for some long rides? Can you take them along on shopping trips? (All the in-and-out of the car for you will give them a chance to get bored, and pretty soon they might be sleeping.) At the least, it will show them that no matter how alarming the noise is outside the carriers, nothing frightening happens to them inside the carrier.
ps -- Buy some incontinence pads, the big, flat kind made for use in a (human) bed. Fold them in thirds absorbent side out, and put them in the bottom of the carrier. They're great if kitty pees, you just roll the pad up and put in another one. Much better than a towel.
no..........I doubt any parent would attempt to guess at the dosage . however , Bringing my cats here to a vet will add $ 600 to this trip and maybe for nothing. They are healthy and yet, you are right about the stress.. this is why I inquired, I know it is stressful. Well, I guess I'll just do the naturopathic remedies but although I haven't had success with them, some other people have..
Please find a Vet you do trust there must be options....you cannot just 'guess' at a correct dose or type of medication that would be appropriate for a cat. A wrong guess could be fatal!!!!
And yes by all means get harnesses for each cat or risk loosing them in a busy airport
Good luck with the flight, very stressful for kitties