I use the Two-Bucket Method: Nestle two big buckets with handles. The bottom one for mop water and the top one for spray bottles and supplies. This is how po folks who don't have rolling wringer buckets do it. I guess you could do the same thing with a wringer bucket and a small bucket sitting in or on it.
I worked for a low-end place once that sent cleaners out in their own cars for several quick jobs around town each day. The routine was to carry a big bag of rags and clean mopheads in a bag with a shoulder strap, carry a canister vac also with a shoulder strap, grab dusters, upright and mop handle in one hand and buckets in another. One person could carry all their stuff in one trip that way.
Speaking of carrying stuff -- I bought a great commercial Hoover upright vacuum that was priced right and does everything I need. The only problem is that it's one thick solid unit below the handle, so there are no handy grabby spots for carrying it around. Didn't even consider that when I was buying it, but the design's a real inconvenience.
I can't find a picture with a side or back view to explain what I am talking about. This is the model:
Any ideas for how to carry it around? The only handle-size part to grab onto is what you can see on the top. It's too wide to grab from the top of the bag on down.