What do I do when I don't trust my LBS to repair a problem they caused?
First thing always is to take it back to that LBS and talk to the manager or owner. At the very least that way you know that they know you're unhappy. Probably they will offer to fix the problem for you (or get one of their mechanics to do it). From working in a bike shop we always took those return/re-repair jobs quite seriously, because one angry ex-customer will tell 100 people but a happy customer might tell 2.

Getting them to pay a competitor to fix your bike is harder, especially if you want to take it there yourself and get reimbursed. Generally they will get a better price than you will just because they're another bike shop, so it's better for them to send the job through. Which they will do if they can't do the work themselves. Small shops often do this with stuff like suspension rebuilds that require specialised tools and knowledge. But for a simple wheel install? Unlikely.

(edit)But now that it's an axle replacement you might have more luck. This is likely to be tricky anyway, since axles for some older SA hubs aren't made any more (AFAIK), so it might come down to who actually has the part. Your best bet might be to find a shop that does deal with older hubs and get a quote from them. Tell them you're going g back to the first shop, so they don't write "replace type 253a axle in SA 3xHD hub" on the quote (because that would let any monkey buy one on eBay). That way you're just doing work you'd have to do anyway, and you might get the actual repair done free. At the very least you have some idea of how hard it's going to be to do the work and what it will cost, for when you're negotiating with original shop.

The question for the shop is whether you be significantly less annoyed if they do this than if they just offer to fix it themselves, and how will that story play out to people you tell it to? Unless you lie you're going to be telling potential customers "they screwed up, and when I took it back they offered to fix it but I wanted to have them pay another shop to fix it and they wouldn't do that". Not something that would put me off using that shop.