Your options are plainly to mix the water with the hot to prevent sweating, or get an insulated tank. At 1.6gpf, you shouldn't be concerned about backups. Cold water discourages mold; warm water is where it breeds, so using a mixing valve is going to enhance your mold problem rather than retard it, but probably not immensely so.
Maybe your plumber likes the Viper, but here we like the Toto Drake. It's available through many suppliers (and online) for $200-ish. Maybe $25-ish more for the insulated tank. I'm talking about in white; colors cost more.
CST744SD is the model number for the elongated bowl with the insulated tank. [pump that in google to see what I'm talking about.] CST744S for the 1.6gpf model or CST744E for the 1.28 gpf watersense-approved model, both without insulated tank.
Look at the reviews above regarding the Drake (not Drake II, that's probably more $$ than you want for this application) [click on green "Toilet Reviews" box above]. It's the best-flushing gravity toilet out there in its price range, it's a premium product with high-quality china and a low defect rate, and simple, American-made Korky-brand parts are readily available now and into the future so when they wear out you can fix it with parts available at Lowe's or the local hardware store.
Changing the toilet should help substantially with your septic tank filling.
Query: do you have good water pressure? Do you have a lot of grit in the water? These things can affect the toilet's fill performance, so I just wanted to be sure.
Come back with questions, comments, and your report.
PS Obviously a bleach-based in-tank bowl cleaner will eliminate your slime problem. but it will destroy the flapper and other rubber parts of your toilet, so it's a conundrum. All warranties are voided by in-tank bowl-cleaners. So, assuming you need a methodology that doesn't involve bowl cleaner: Have the plumber give you a quarter-turn ball stop on your toilet water supply if you don't have one already, and get yourself one of those piston-like toilet tank water-suckers (look up "Last Drop Suction Gun" online and you'll see what I mean; you can get it for about $20; amazon has it from some dufus for $120; don't do that.). That way, when you leave for the week, quick 1/4-turn of the stop valve on the water supply, flush the toilet, use the piston thingy to suck remaining water out of tank, leave lid off, and it should be in good shape when you return. (Or, if you're of a different mindset, buy the Drake, get yourself a supply of 4 or 5 Korky "Fits Toto" Flappers at Lowes for $10 each and keep them in the drawer at the house, along with at least one Korky 528MPK Max Performance fill valve from Lowe's, and maybe a couple of good tank-to-bowl washers, and use the bleach tabs to your heart's content, just replacing the flapper when it starts leaking... Of course, because the flapper WILL start to leak with this setup, you probably want to kill the water at the toilet when you leave for the week so you don't have a runaway toilet between visits. By the way, this isn't a completely-insane idea; there was a recent post I read from someone who decided that they couldn't live without bleach-based bowl cleaner and bought a supply of the Korky flappers because it's the most-resistant to chlorine, and she just replaces the flapper as needed. She may be less happy when it comes time to replace the flush-valve gasket and/or the tank-to-bowl washers, because those involve removing the tank...)
PPS Gary, who is a real expert, makes a good point below about low-flow toilets being very resistant to sweating. The reason that I'm thinking you are probably one of those who actually might need an insulated tank or mixing valve is that I am imagining a group of people doing activities together, returning to the vacation cabin together, and lining up to use the bathroom one after the other, maybe flushing a couple of times each. In other words, maybe ten flushes in a half-hour. If your tank is sweating so much now that you need a system to catch the drips, I'm thinking that there's a lot of moisture in the air and that your usage patterns plus very cold well-water may combine, even with a low-flow, to cause the tank to sweat.