all those "not an options" can all suddenly become required if the stand or tank fails. a wobbly stand puts extra pressure on the tank seals so they rupture. better to do the right fix now than multiple repairs later.
ruptured seals, especially near the bottom, means you'll probably need to pull up your carpeting anyway to avoid mold. wouldn't you rather pull up dry instead of fishy wet carpet?
you need to define your wobble. if it's just some give under the stand, you could be ok. the little bit of wobble from carpet compression/decompression will fade some as time goes on and the carpet fibers flatten as it wobbles. a simple bump probably wouldn't do it, but repeated rocking might send it over. i.e. an earthquake or small child pushing it back and forth to watch the waves.
if your stand is flexing, you have a problem. you may be able to add braces inside the stand, maybe not. if this is the case, and a new stand is truly not an option, I'd break down the tank. too much disaster potential.
it may even be a bit of both, especially if it's a cheapy pre-built stand.