To start, I would second that idea of dropping a tube into that tire, as "JL" suggested above, even though it is new, or return it, and put the old one back on with a tube. Never been much of a believer in plastic rims.
Here's a little trick I used just yesterday on my old Scott's lawn tractor. I could not get the bead to pop in on one side of the tire for love nor money. The tire does have a tube in it however.
So, I deflated the tire sufficiently to be able to get my finger in between the bead and where it supposed to sit on the wheel. I slathered cooking oil over both surfaces until it was a slippery mess.
I then positioned the tractor in the Sun so the tire would warm up for an hour or so. (Make sure the tire sidewall is nice and clean. The blacker the surface the more heat it will absorb). Then I re-inflated the tire to just under maximum pressure.
Then I drove the tractor around over some rough terrain where the tire would be subjected to various types of compression and distortion, which would affect its shape. Within about two minutes of that, I heard the bead pop, (even over the sound of the engine), and sure enough, the tire was seated properly.
This worked, partially because the tractor is heavy and will distort the tire over bumpy surfaces. Remember, even though sometimes they can be a real sob, that tire really wants
to sit on the rim properly, but occasionally they take a little "convincing".
You might be able to do something similar, but make sure you've got some weight in the wheelbarrow like a couple of cinder blocks or something. Then just keep going over the bumpy terrain and maybe you'll get lucky.
The worst that can happen is that you'll get a little tired, and of course, if your neighbors see you running around the yard in circles with the cinder blocks they'll probably laugh at you, but most of us here are snowblower collectors, and we're used to that ....