and I think I need to soak my hands in the bath for an hour.

That doesn't bear thinking about

I think it might take more than an hour in the bath to get rid of the smell - perhaps you ought to use some form of barrier cream.
My boy suffers quite badly with smegma buildup and I wouldn't dare go in without a pair of gloves, even then the smell lingers - if you know what I mean

One of the others has a real phobia about being anywhere near his manhood, and has needed sedating in the past to get him to drop and have it clean, but I swear this has only made it worse. He really objected under sedation and even had ACP which didn't help much either.
I have to admit I hadn't thought about clicker training to try to help with the issue, but I will definitely be giving it a go to see if it helps.