the nut doesn’t have to thread so far down on the axle, that you have axle sticking out past the top of the nut, when you’re dealing with captive nuts. Outside width of the nut is irrelevant. What’s relevant is the height of the captive washer and nut together. The mechanic at the shop will be able to determine if your axle has enough threading to safely engage the nuts they have. This would be a completely different scenario with non-captive nuts. Typically non-captive are taller (require more threading) than captive or ‘track’ nuts. Precisely because the captive washer means less threading is required to properly secure the nuts. So if your bike currently has non-captive nuts, that have enough threading, you should have plenty of axle threading for track nuts.

while you’re at the shop, ask the mecahic to let you tighten down a nut with a standard size wrench, to your strength ability and see if that’s secure enough. I’m willing to bet, you’ll be able to do it fine.