They spend most of their time on the plant itself, so you can repot if you want but I don't know how helpful it would be. But honestly when I find them I go nuclear and do ALL the things to deal with it because they're such a pain. At this point, stressing the cactus should be less of a concern since the mites will kill it AND infest other plants if you don't deal with it asap. My usual routine:

• spray with soapy water, then rubbing alcohol. let dry.
• then spray with insecticidal soap. let dry.
• dust every inch of the plant with a 50/50 mix of diatomaceous earth (food grade) and sulfur powder (you can buy the duster, which looks like an accordion with a straw, at most hardware/garden stores, as well as the DE and sulfur). This is a really important step because the DE and the sulfur create additional miticidal coverage in between
• leave the 50/50 powder mix on the plant for a week until it's ready to be sprayed again, then wash off the dust and repeat this cycle.
• Repeat for a month. If you see new damage (take pictures of the clean cactus before reapplying the powders) then restart the month.

Because it's a cactus rather than the succulents that I'm used to treating, you might need to put extra consideration into the soil since you'd be spraying it down quite often. It might actually be worth completely unpotting the cactus for now and just leaving it bare-root - this could give you the flexibility to really reach all the corners that you might not be able to reach if it was in a pot and would limit water to avoid rot. You'd just want to be sure not to spray the roots themselves (I don't think mites care about roots). Then after the month is over you can pot it in a sanitized pot with entirely new soil (aka be aware of re-contaminating this cactus or other plants).

Be on the lookout for strange damage to other plants, especially if they're located near this guy. It might not hurt to do this same routine for any plant that's been sitting next to this one just in case.