For most brews you should be able to use an airlock without fouling. I use a 7.9-gallon fermenter for 5 gallons, and there has been just a little crud it the airlock one time out of over 100 batches using it.
I would also have some concern about the bucket material. Maybe with some effort you could find out if it is food grade.
I have two 7.9-gallon buckets from BSG Handcraft. Have had them for about a year; have never used them. There is a ridge inside at the six-gallon level. The actual lip of the bucket is only about two inches about that ridge. In my inexperienced opinion, so little headspace is just asking for the krausen to get into the airlock. We must remember the lid protrudes into the bucket by maybe 3/4-inch, further reducing the headspace.
I'm going to ferment 7.5 gallons of something in the garbage can with towel & plywood top, drain-off 6.0 gallons of trub-free deliciousness into my bottling bucket, bottle and cap, then wait an agonizing 30 days before cracking the first one. I'll be eyeballin' the bottles for signs of infection. I have consumed infected beer in the past; don't remember if it was unpalatable or not. Because I cannot remember, maybe it was drinkable enough. That would have been back between 1990 to about 2001. After that, I got too busy working and didn't make any beer for close to twenty years.
If the garbage can method works well enough, I'll start making my grog that way and to helsinki with the fermentation buckets I have. I want to bottle six gallons; that's the goal. It irks me that I lose a gallon to the dead yeast in a 6.5-gallon bucket. My solution is to ferment more than I want and discard the rest. Yes, that costs a bit of money... but I do not want to be fighting to keep the dead yeast out of my bottling bucket. I have no wife nor kids, so I can spend the money required to make the beer that gets poured down the drain instead of worrying myself all to pieces about the yucky yeast crap getting into my bottling bucket and ruining the flavor of what I'll drink a month from Bottling Day. Sometimes you have to lose a little to gain a lot.
I'm not going to stress over what is the material of the bucket. It's white and flexible, so I'm going to assume it's either food-grade or some kind of plastic that won't kill me dead if I don't drink 13.5 gallons of water out of it on a daily basis. I expect the manufacturer didn't bother to label if/if not the can is food-grade because I doubt he'd ever expect anyone to consume for food or drink something coming out of a garbage can.