Well.. Personally, I wouldn't dare clean anything within ~2 ft of a broody button - and if she seemed bothered, I wouldn't even do that. But it might not be a problem, I don't know. Basically - if you'd wish you hadn't done it if she ends up abandoning the nest, then you probably shouldn't do it. But if the cage would get so seriously nasty in another 16+ days, that missing a chance of button chicks is better, then so be it. Perhaps you can just add a bit of extra litter on top of the dirty litter, without getting too close to the nest? Or, if the cleaning can be done very quickly, perhaps clean the area near her nest when she's left it to feed, to avoid stressing her.
I also have a button hen that I think has just become broody and I certainly don't intend to clean her cage before she has chicks, but she lives in my room so it's relatively warm and their poop dries quickly so it doesn't really become 'nasty' in there even if I don't clean it for a long time.