By the way, you don't have a Rialto, you have a Rosario K-3434. It uses exactly the same flapper on that square flush valve as does the Rialto, however. It is Kohler part no. 83064, which is what the Korky 2010 replaces. I would get the Korky.
The Rialto has a whole different fill valve setup, which is a pain to replace or repair; has a rim flush thingy and such. The Rosario just uses the standard universal Kohler replacement fill valve, GP1138930, which is basically a Fluidmaster 400A, so I'm a little surprised by the two fill hoses. Maybe HJ or someone really knowledgeable would have a clue why yours seems to be set up that way. I don't know what fill valve you have in there -- it's plainly a 400A variant, but since the toilet uses the standard one-hose replacement fill valve, you could replace that with a standard 20-percent-refill Fluidmaster 400A or Korky 528 (the cheapo one with the white top), and have just one fill hose.
As to the gasket, it's Kohler part no. GP83888, which isn't green, so it appears that some work has been done on your toilet since it was installed (which may explain that fill valve). The gasket connects between the flush valve assembly and the tank. If it ain't broke, i.e. not leaking, I wouldn't screw with it. However, if it's leaking and you want to replace it, just Google Kohler 83888 and you'll find it and/or an equivalent offered on a lot of online mechants. I haven't worked on a Rosario, so this is just informed conjecture, but what you do is (turn off the water and drain the tank and) push down on the whole flush valve assembly and turn it 1/3 turn in either direction (I think yours will only turn enough in one direction the way it's in there), then pull the whole assembly and that gasket up and out. Clean the toilet tank floor where it's connecting, clean the requisite area of the flush valve assembly, put the new gasket around the flush valve assembly, put it back in the hole, push down hard and turn it 1/3 turn until it stops. Done. This is easier said than done without issues, though, so unless you're the adventurous type with some time to kill (which I sadly am), I would leave well enough alone if the flapper is the only issue. Me, I would change that fill valve, though, and would probably try a Korky 528 since it's so easy to install.
PS Looking now at your second photo, I can see the 3434 embossed in the tank, but I can't tell which version it is [it would be -AA -AB -BB or -DB], although it is at least since the 1/1/94 implementation nationwide of the 1.6gpf rules -- it has 1.6gpf embossed in it as well...