Let's learn about energy: BTUs.
Put 1 pound of water (just shy of a pint) on the stove and stick a thermometer in it. Turn the stove on, and watch the water warm up on the thermometer.
You're watching energy in action. Every 1 degree of temperature rise means you added 1 "BTU" (British Thermal Unit) of energy into the water. So if it's 62F and you're going to 212F, it takes - can you guess? 150 BTUs to raise that pound of water 150 degrees.
And time this. You can figure out how many BTUs per minute your stove can input into the water (at that flame setting). Right?
So how many BTUs does it take to get to 213 degrees, i.e. boiled off water vapor? It's only one more degree. One more BTU, right? Nope.
In fact you can time it. You know how many BTUs per minute the stove puts out. How many minutes to boil off all the water? That will tell you how many BTUs it takes. It's about 1000 BTUs. That converts water from 212F liquid to 212F vapor.
What's the deal with that? That is called the Latent heat of vaporization aka latent heat of enthalpy. It's the energy required for the state change from liquid to gas. You have to pump that 1000 BTU into the hot water to make it boil.
By the way, when a heater or A/C specifies BTUs, it really means BTUs-per-hour. This is a rate unit, indicating power (like watts) not energy. A true BTU is what you just did on the stove.
And... Propane has that too
Think about how liquid propane works. It's delivered as a liquid. It sits in the tank as a liquid. But you use it as a gas. So what's happening? When you open the valve, some of the propane boils to produce the gas you are using.
But wait. Doesn't it take a bunch of energy to convert from liquid to gas? Like 1000 BTU per pound? Yeah. Yeah it does. About 1/5 of water: 184 BTU/lb which is still significant. So where does that energy come from?
The propane absorbs it from the propane in the tank. So if you started with 80 degrees F liquid propane (say it's summer), you soon have 70 degrees F liquid propane. Then 60 degrees F propane as you use more and more... 50 degrees F... 40 degrees F... and by the way, you just discovered how air conditioning works :)
Meanwhile, the tank is being warmed by the environment. Heat transfer rate is decided by the difference in temperatures (propane vs outside). In the summer, that's easy - A 40 degree difference (80F vs 40F) absorbs heat twice as fast as 80F vs 60F. It reaches a balance at one point or another. In the summer.
But when you do it in winter, say it's 0F outside... you run the propane, and it gets colder... -20... -40... Now your temp difference is 20 degrees... then 40 degrees and heat transfer doubles... Except, something happens at -40. That is the boiling point of propane naturally (like water's 212F). Below this temp, it doesn't want to boil at all. So what comes out the pipe is either liquid propane (which you Do Not Want) or far too little propane. Either way, the propane supply has broken down.
And because the outside temperature is so low, the propane tank has very little source for heat. As a result, it can produce very little gas.
But of course, that's exactly when your demand is the highest, isn't it? Your furnace is running hard, the generator is working hard, you're cooking, using hot water...
Fixing it
As you discovered, you can put warm water on the tank, and that will temporarily inject some heat, to get the propane able to boil more gas. But it certainly won't last. Worse, that propane tank is going to freeze any water that touches it, so additional water will just be wasted, as it won't even melt the ice that has crusted onto the tank from previous pours. So that strategy won't really extend.
Your best bet, short term, is simply to reduce draw. Reduce cooking, hot water use and generator load. Can't do much about furnace load obviously. You could even put a thermometer on the tank proper, and just know to cut your gas usage when the tank gets too cold.
You can also try to keep the tank topped up. When the propane is only touching the bottom 1/4 of the tank, that means heat is only transferring through 1/4 of the contact area with the outside. (Plus a little bit of conduction down the steel tank sides). More propane means more thermal mass in the propane, too, which helps with short term bursts of usage.
Keep snow cleared from around the tank. The bottom of the tank is the most reliable surface since it works at all fill levels - you can't afford to have it out of commission. You want the wind to whip freely across the underside of the tank; if it's blocked by snow that takes away much of your heat source when it's near empty.
It seems super weird to see 0F or -10F ambient air as a heat source, but for keeping a propane tank above -40, it does work.
You can try heating the tank. For instance they make heat tape designed to prevent water pipes from freezing; you could wrap it around the tank, focusing on the bottom. Or you can plumb the generator's antifreeze out there on a couple of insulated pipes, and have some steel tubing strapped to the bottom of the tank to warm it. Just make sure it is a low-heat source that does not exceed about 200 degrees F and has no open flame. It might be good to discuss options with your propane supplier.
You could also try strapping "cooling fins" onto the tank; the goal being to absorb heat rather than radiate it. You want to increase the thermal surface area of the (bottom of the) tank as much as possible.
You can also increase surface area by getting a larger tank. You want a thin, narrow tank, not a short plump one, to maximize surface area.
Don't insulate, much
And you don't want to insulate the tank; at least, don't insulate any more than the area around the heater. The steel tank's free contact with the air is how it works normally.
Some tank heating solutions insulate the whole tank. But this is due to a wrongheaded notion of the problem, and the effect is to make you 100% reliant on the heating solution! If it breaks down, your propane delivery fails because the tank can no longer absorb heat from ambient.
Let's follow the bouncing ball. We need to keep propane temp well above -40. Suppose ambient air is 0F. We use a system that heats 5% of the tank's surface to 150F, giving very good heat transfer, and that immediate area is insulated so our 150F source doesn't heat the world. With convection (self-stirring of the liquid) we get the propane warmed up to -5F even while gas is being drawn. So we are holding even at -5. The worry is "oh no! With only 5% of the tank insulated, it's heating the world through the uninsulated 95% of the tank!" Actually, no, it's not.. The propane at -5F is colder than the outside at 0F, so heat is moving from the outside into the tank still. Outside is helping, not stealing. Of course, a warming system could install some intelligence to ensure the heater never runs unnecessarily, by comparing internal and outside temperatures, and only operating when outside absorption is insufficient.
Insulation only makes sense when you're dealing with extremely bitter Siberia/Valdez/Nuvanut cold, where ambient air being -20 or lower is a regular thing.
