You definitely need a bigger bucket. A 7 or 8 gallon one would be ideal. To my mind, a six gallon one is just too small, especially for a brew like that where you're really giving the yeasties a feast (actually, a six gal bucket for even a normal 5.25 gal of beer can be pushing one's luck a bit, if the yeasties are really active).
In the future you could install a blowoff and avoid the mopping (but still potentially lose a lot of beer...not worth it in my book).
I still maintain that a bigger fermentor is the best solution. I'm still using a 7 gal glass acid carboy for 5 gal batches and have had only one blowout in 23 years (and it was my fault...I overpitched a fresh yeast cake a bit too massively). Eventually though, I will probably switch back to a bucket sooner or later. I've seen some very nice 7 and 8 gal brew buckets, even some very translucent ones, that would cover the problem easily.