Shim the stand for sure. I would not shim the tank (too dangerous in terms of stress), nor would I cut down the stand (permanent, and might alter structural integrity.)
Take your time to measure the exact height needed. <Use temporary shims to get it perfectly level, then measure.> Then make a wood perimeter stripe (all 4 sides, assuming the stand is resting on the floor on all sides & not raised, plus any extra supports the stand needs in the middle) that is the exact profile needed (lots of planing & sanding!). Use maybe a 3" wide stripe for ease of working (or as wide as the stand's support), and make it so it is just proud of the edge of the stand. <May be round off the edge for a nice profile, miter the corners so it all fits.> Then turn the stand over & nail the stripes to the base of the stand, finish it to match the stand.
You are basically making a low profile platform (that is tapered to fit your floor) for the stand to sit on. (I am thinking maybe 3/4 -1" so the difference of 5/16" won't be too noticeable.) Since it is all the way around the stand, it will look like part of the stand & not a hammered in shim job.
I hope I am making myself clear!