You don't need a new tire for a slow leak. As rhagfo says, it could just be a bad valve. If it's the tire, a plug or patch will make it like new. If it's the wheel, you want to know that.
Doesn't it have a spare? I'd put that spare on it and take the defective one to the tire shop -- or just pull the whole rig to the tire shop again.
If you do have to get a new tire (unlikely) I'd get the same kind that's on it. If you need one now for a trip, see if you can't find an old one for a spare just to give you some time for the new one to arrive.
I've usually used my own muscles and sweat to change tires, but I don't hesitate to call my insurance company for roadside services either. We pay for that insurance, and it doesn't drive up your rates like an accident does.
I've never had a trailer that came with a jack and tire wrench. I don't know why they don't. I bought my hydraulic jack 45 years ago. It's gotten me through many trucks and campers. Mine cost more in '68 than new ones from Walmart do now.
Be sure to get (make) some platforms for the jack too, so you have enough lift to get the tire off the ground. I have a couple 2x12x12s nailed together and another couple not nailed that I also use for stabilizer platforms.