Please clarify which protein you want to work with, the SUMO-tagged protein or the supernatant after cutting off the SUMO tag. I suggest you try them both and compare. Please be aware that some journals may not accept enzymological data from the protein with the SUMO tag still attached.

The supernatant contained my protein, and when comparing before and after spinning in the gel, I had only lost around 50% of my total protein amount. When I loaded my supernatant (containing soluble protein) onto gel filtration, it runs as a monomer. However, when concentrating the relevant fractions for storage, my protein crashed out again in 0.3% CHAPS (suggesting a concentration-dependent aggregation behaviour). When doing gel filtration in the same buffer but with 0.8% CHAPS instead, the protein also runs as a monomer, and when concentrating it did not crash out. How far would you trust the monomeric protein in 0.8% CHAPS? Do you think at this concentration CHAPS in preventing aggregation, or disrupting the protein structure? Personally I have seen people using up to 1.5% CHAPS in their lysis buffer, so dont think it's too high

One way to deal with this is to do assays with several different concentrations of CHAPS. If it has no effect other than increasing solubility, you're fine. Otherwise, you at least know you have a problem. The same applies to the tag: if you can show it to have no effect on enzymatic properties, you are free to use it. In both cases, you evidence must be good enough to convince not only yourself, but also the reviewer of your paper.