A jigsaw ain't gonna git 'er done. A Sawzall and a whole buncha demolition blades will be slightly better. You're going to have to look hard for the blades too because most of the blades out there are "bi-metal" which does not include ferrous metals, as you'll see in the fine print. And you'll probably find it easier, as in "remotely feasible" to cut a square or rectangular opening versus a circular one.
Nope, torches don't cut stainless for beans, either. What you really need is a plasma cutter. OK, that's out the window since the first place you and I both would look for one of those is the local emergency room. (g)
I've cut a bit of stainless, of varying thicknesses, and what I'd recommend you use is a hand grinder with what is called a cut-off wheel. It looks sort of like a grinding wheel but is quite thin and designed for cutting, not grinding. And you're going to end up cutting a square or a rectangle with it.
You'll need safety googles, a long sleeve shirt, and gloves. You're also going to want to have a bunch of wet towels to place over things that the large amount of sparks coming off the wheel are otherwise going to burn or melt, like fiberglas.
I think, once access is achieved, that I might be more inclined to epoxy the inside of the tank rather than patch it with caulking. And I'd probably do all of the seams, maybe the whole inside, while I was there.
You can get some stainless "cut-offs" at your local metal supply house to fabricate covers for your holes once you're done. Some eighth inch rubber gasket material will be needed also. Given the joys of working with stainless, you might find it easier to just glue those on with the 5200 instead of fabricating for bolts, etc... By that point in the operation, chiseling them off at some undetermined later date will hold little terror for you.
By the way, water really only serves as a coolant for things like cutting concrete, etc...it's not going to lube anything.
btw, I think you'll go through about the same number of hole saws that you would sawzall blades using those for a round hole and they'd end up costing about what a new tank would by the time you were done.