If you don't want to install underfloor heating mats for the porcelain tile (which I would do, but raises the price) then the vinyl will be your best choice. The stained concrete is an expensive choice, usually amounting to 2x the amount of installed tile. With you investing in finishing the basement, you need to be sure that any moisture issues are addressed first by double checking the key mitigation factors such as proper guttering, overhangs, and the slope away from the house. You can still have all of those things and still have a moisture issue, so again, before you spend money to finish things, make sure that you don't have those issues.

You are welcome to put the vinyl over top the marble. Be aware that you will NOT be able to do the preparation needed for the vinyl to be properly installed. The preparation means you have to a SMOOTH and FLAT subfloor (grout lines would, ideally, be level with the tile) and any humps/bumps ground down/flattened and any hollows filled with concrete. Ahhhhhhh. Now you see where this is getting tricky. To "cover" the marble (to "save it") you may have to destroy it to make the vinyl install "proper". If you do NOT do the proper preparation, you will get a vinyl floor that will telegraph ALL THE GROUT LINES through to the surface. You will end up with a "checker board" pattern of grout lines showing through the surface of your newly laid vinyl. Here is my suggestion: Water-Proof LAMINATE! Yep. Something like Aquagard (at several dollars more per square foot) will do it. Why? Aquaguard is RIGID. And it allows UNDERLAYMENT which can help with any minor imperfections. Yes...technically laminate wants a properly prepared subfloor - but it is more forgiving visually. You might get a snick of movement over hollows but I would take a tiny bit of movement over a grid pattern etched into vinyl any day. As for the baseboards....remove them and install something that fits your aesthetics. The one's visible are nothing special. Some of them don't even match the tile they are attached to. Not worth saving in my opinion. Have a look at water proof laminate. It offers the same wearability without the issues with excess heat/direct sunlight and without the issues of a super flat subfloor.

Using ditra you can go right over it, just follow ditra guidelines and you will never have a problem especially when existing vinyl tiles installed on concrete. Keep in mind that Ditra allows dissimilar materials to act independently of each other this why the tile or grout will never crack even if there is differential movement between the subfloor and the tile. We used ditra over asbestos tiles installed in the kitchen about 6 yrs ago over an old wood floor in a house that was built in the 20s and there was no issues then and there is no issues a year ago when I was there doing a deck repair. Good luck

The cost of a PROPERLY laid micro-topping that is then polished/ground to a high-level finish will depend on whether or not the current slab has an existing/functional moisture barrier. Some of the SUPER HIGH END microtoppings that are DESIGNED to be polished (no epoxy coating) to a super-dense finish, are very expensive. They can take close to 2 weeks to grind down the existing slab, then put down a moisture barrier (spray-in-place x 2 coats), then primed, then the micro-topping is put in and then all the grinding to get a closed-pore floor that is almost impervious to stains and etching (almost...but not quite). And a professional concrete flooring specialist isn't cheap. The level of "polish" will determine your costs. For a DIY level finish = $5/sf (which stains and etches and takes ++maintenance and ++ polish/wax). A medium grade finish (professional level) WITHOUT the microtopping = $7-$10/sf. A high-end (almost bullet proof finish) which includes the microtopping = $15 - $20/sf. And then the actual costs will depend on your location and the ease of access to all of the products/professionals you need. If you are rural without access to a large city, then the cost is going to go up. If you are in an urban location with LOW wages, then you are set for the lower range that I've listed. If you are in an EXPENSIVE urban location (think NYC) then you are in for the HIGHEST range I've listed. But until you decide which type of FINISH you are after, we can't help much more than that.