Thanks everybody for your help!
After doing the research, it turns out that the difficulty isn’t in finding a bus company that lets you bring pets on board. Aloha, Ho-hsin, Howtai, and many more all say that animals aren’t a problem. All companies ask you to crate your pet, and most of them ask you to pay half fare (though pets are not assigned a seat, so they’re crammed in with you). The service lady on the phone at Ho-hsin told me I had to muzzle my dog AND cage him, but other people who have taken their pets on Ho-hsin have not reported having to muzzle their pet.
I’d originally written GuoGuang off the list because their rules said all pets must be kept in the poorly ventilated cargo area, so they did not recommend bringing pets. Dragonair also says no pets on their website, and out of consideration for the other riders, no pets above deck, not even in a cage.
But after hearing about and reading other people’s experiences, it seems like the rules are flexible. Small dog owners have reported being allowed to let their pets sit in purses instead of cages. If it’s an empty bus and you run into a nice driver, the rules are more easily bent.
We finally went with Guoguang just because they seem to be the only freaking bus company with direct Taipei-Puli routes. I checked three times before purchasing my tickets – once on the phone, once at the information desk, and once with the ticketseller – and they are assuring me that I don’t need to keep the dog in the cargo area. Again, I hope the rules don’t suddenly change when it’s time to board the bus! I paid half-fare and asked to be seated in the back, where there is slightly more room to put his carrier on the floor, and they were willing to accomodate that request. I told them I was willing to pay full fare for the dog if it meant we could guarantee a spot for him (and yes, he would have remained in his cage), but the woman snorted and just repeated, “Dogs are not permitted to occupy seats.”
It seems like this was a huge pain in the ass for a short weekend trip, but hey… this was information I would have needed, sooner or later. Being scooter-less, car-less, and virtually friend-less, I’ve got to rely on things like mass transportation. 
