Build a collar and you won't need to worry about the bucket getting nasty... OR, change it after the first 3-7 days and cover it... I typically use one of my spare 2 gallon fermenting buckets (for 1 gallon batches of mead, originally, now just ways to measure water and for blow-off assemblies)... If you use something else, then get a way to have the opening closed better (if a bucket) and go from there.
I plan on making a fermentation chamber from a freezer at some point too... Most of my fermenting, by that time, will be in kegs. So, chances are, I'll need to build a collar for the chamber. IMO, it's well worth the effort to make the collar, since you'll expand how much you can ferment inside the chamber...