We regularly airfreight drums to the USA. Drums are packaged individually in PR AAA rigid cases & further packaged. That's an expensive way of doing things, but we do it for reasons that don't apply to sending stuff privately.
Nesting the drums & shipping by sea is the most economical option if you're not in a hurry. You'll need to make a declaration to avoid importation duties, but that's not difficult (unless there's banned wood species or something similar involved).
Packed individually & cased, by priority air, you're looking at around £250. By sea & nested, you should be looking at under £100 door to door, or less if you can deliver to a hub & pick up yourself from the port. Remember, part of the cost is taken up with export / import document processing. It's not something you can reasonably sort out yourself, so you end up paying for that service either included with courier fees, or by paying an agent. Also remember, you should consider insurance, & that can add considerably to costs.
If you're thinking of eventually bringing the kit back to the UK, then there's various temporary export mechanisms you can choose from, but they're all a PITA, & require record keeping / chain discipline. Unless you're thinking of leaving the kit in the USA, or this specific kit is important to have with you, I'd strongly consider leaving my kit in the UK & buying something cheap in the USA. If you buy right, you can probably recover most of your cost on resale.