Major Payne is the hardest one to beat, hands-down. Because of the Bear.
The usual cookie-cutter answers to this are:
"You just need a Fluxfire Feline." or insert perfect counter.
I think these answers fall short because no one should have to tame a specific (and overpowered to boot) pet just to counter something PvE. If something has only one counter, it's not a strategy game, as pokemon is supposed to be. It's rock-paper-scissors, with the emphasis on scissors just because it's made of metal and gets to be OP and tear up rock too.
With over five hundred battle pets in the game, there is no excuse to be forced to have a raised Fluxfire. ...
FFF is going to be nerfed soon anyway, so giving that as an ideal counter was a bit silly.
It is helpful to use an appropriate counter but there are whole families to choose from, not just one or two pets. I did all the trainers, including spirit tamers, before I had any of the recommended pets. I still don't have Anubisath Idol and have never used a Flayer or any other humanoid against the trainers.
I recommend getting rare breed 4s of Shimmershell Snail and Wildhammer Gryphon Hatchling, these 2 are easy to farm and are in most of my trainer teams. Good alternatives are any of the water striders/skimmers and a stoned Yellow Moth, (highest attack of any of the moths). My favourite mech pet is Darkmoon Zeppelin, well worth getting if you feel up to collecting 90 prize tickets.
Don't feel you have to rush through the trainer quests. Take your time and have a bit of fun catching and levelling pets in your spare time. The trainers will wait, and one day you'll find them all (usually) easy.