One thing that irks me more than dead cell phone battery or a floor repair is a sweaty toilet tank. I've read lots of fixes for these, and I'm going to give you mine. You know your situation best, but I live in a mobile home and some of what works for a stick built home will be a pain for a mobile home. For one thing, more fittings in a stick built home are bad, but in a mobile home, these can be disastrous. Water leaks in a mobile home are as bad as a fire and just as destructive. Installing a tempering valve will take away from the aesthetics of the bathroom. Mounting the valve below the floor or in a cabinet will be inconvenient or impossible. If this is one of the old school toilets with a 3 to 5 gallon tank, part of the problem is that the water is going to stay cold for a long time. Trying to temper this with hot water will add up after awhile, so unless you happen to have stock in your utility or propane company, I cannot recommend these valves for a toilet. If this works for your abode, then you don't need my advice. These valves can fail, are expensive to install and if you have an older toilet are not going to give you the results you're looking for.
For once I'm going to strongly suggest you replace the entire toilet with a newer, low flow model with an insulated tank. Insulation kits for a retrofit will not last even if you do manage to get them to stick and more likely than not, will not work. I am going to add that a good toilet that you won't need to plunge every other flush isn't going to be cheap. The $50 to $60 setups will clog to the point of driving you to distraction. Some of the ones that use air pressure will work, but you will pay dearly for them. In excess of $200 or $300. Niagara toilets are the best ones on so many levels, but these are also in the upper $100s if you can even find one retail. These are sold wholesale only, which isn't an option. However, there is an alternative that will work to give you a good flush, stop the sweating, and look good.
Menards is one home center I don't mind going to for my home improvement needs, some work tools, and maybe even some clothing and grocery items. They have a huge selection of toilets online and quite a few in store. The one I'm going to point you out to, as long as you don't mind white, is made by EnviroFlush for under $120. It's white, contemporary décor with an elongated bowl and ADA approved. I comes with everything short of a water line to install it and the directions are very clear and concise. It has a soft closing seat, which is cheap looking, but very durable and most importantly, an insulated tank. For those environmentally conscious, this beauty uses 1.28 gallons a flush and works very well for the money. It beats our 1.6 gallon American Standard we spent $140 on a couple years ago in the other bath when the tank on the old one cracked. We have to keep a plunger in that room all the time. With the EnviroFlush, we've never used a plunger, ever. The tank has never sweat it out once either.
Because the directions are so good, there are only a couple things to add. You need to replace the stop valve when you replace the toilet if the condition is at all questionable. If you don't have one, now is the time. Spend the extra few bucks and get a good quarter turn one as these will last nearly forever. The old school twist and shout ones will fail in a major way when there's a leak and the main shutoff is so far away. Add a flood guard line to this for good measure. Your insurance agent will thank you. I receive no money for products I recommend. Maranatha!