I've never done it with annual sweet peas, but I have transplanted perennial sweet pea (Lathyrus latifolius). They survive the transplant well. If it is the annual type you have, I suspect they will transplant fairly easily, just like many other vigorous annuals.

OR, what about some lattice in your whiskey barrel? I do this with climbers, and I like the effect. I've got some containerized cucumbers and beans climbing up a support right now.

Cheers!

I started my sweet peas in pots and transplanted them shortly after they had their true leaves. They transplanted well but stalled for a bit and are now just taking off. I still have a batch of others in pots that were planted the same time and still waiting to be transplanted. They are far more healthy looking and bushier than the ones I transplanted. Perhaps because they have been babied.
Just take care when transplanting and if you have slugs in your area protect your transplants as soon as they're in their new spot.
Keeping them in the whiskey barrel and adding something for them to climb would probably be the best idea. Why risk losing them at this point. Start another batch and put them where you want them. You'll have blossoms a little later to enjoy.

you could also make a teepee out of tree branches or something. It worked well for me. filix

Don't put then in shade. It would be better for them to sprawl on the ground then to shade them. Just put in a few stakes. Peas do not require a heavy duty trellis , just something that can take the wind and may variates don't need much more then 12-24 inches and depending on spacing will support each other. But again shade will starve your plants rather badly. Most vegetable plants are full sun especially fruiting types.