Skip this if you only want to hear about the heat treat
So, the ring lord says that their carbon steel rings are 1062 carbon steel (mostly iron with .62% +- 0.05% carbon). Most 10xx series steels below 1095 have a fairly simple heat treat, but first take a moment to experiment a bit. When you heat up steel you get colors depending on how hot the steel was due to oxidation types. these are a reasonable way to judge how hot your steel got. If only the area directly around the weld is blue/washed out light gray I would consider it fine as it should retain much of its integrity. The higher temperature the steel gets, the more flexible, and soft the steel gets.
However with temperatures in welding you run into a problem called "grain growth" which is farther into the science of steel than i'm gonna get, but makes the steel brittle.
For knives/swords many other people (and I) use a 400 degree (F) temper, but this is fairly brittle for armor. TLR website does not state what temperature they temper to and only give us a KSI rating. You may not get the majority of the steel hotter than TLR tempers, but thats a lot of "if".
However, as a steel geek I digress, back to the heat treat.
Actual heat treatment info
For heat treating 1062 you are going to need two important things. A home heat treating setup, or the phone number of a company or person who does. I would recommend doing some research into local companies that might be able to heat treat for you, but if you are like me and unwilling to pay someone else to do it you can make a heat treat setup yourself.
For heat treating 1062 you will need to get the whole piece up to around 1475 degrees (F) and hold it at that temperature for ~5 minutes at minimum. You can hold it at that temperature for up to ~15 minutes, but thats more up to what you feel like doing that day.
Once it has gotten to that point you will remove it from the heat and quench in it in lukewarm water until it has stopped boiling the water. After this it will be glass hard and can shatter if you drop it on the ground. I would then put it in the oven at 600 degrees (F) for an hour, then let it cool.
That should give you armor hardness, but flexible enough not to shatter if dropped
With anything involving lots of fire, this is dangerous. This is very dangerous. I have been heat treating things for multiple years, but still burn myself every once in a while. you also do not want to be known as that one dude who burned down their block trying to make a cool chainmaille project.
Almost any heat treat will be risky. Even with a fairly simple steel, and a fairly thick piece (rings), and a disregard for how perfectly straight your piece has to be coming out of quench, there is still a chance it will go wrong. There is a chance some of the rings might crack, or shatter, or warp so badly that you have to remove them and put a new ring in.
There is also the chance that nothing will happen. Any number of things can make it so that your piece will not harden at all. A way to check would be take a metal file and scrape it along the rings after quench. If the file skates as if it were on glass, then the steel is hard and super brittle until you temper it in the oven.
There is a chance that you will decarburize the outer layer of the rings and then end up with a thousandth of an inch around every ring that is soft. This means you will have a scratch prone exterior that will eventually wear away leaving hard steel below it (you can remove this yourself though various methods like putting it in a barrel of sand a shaking it a ton)
There is also a reasonable chance I might be wrong about parts of this. I have never heat treated chainmaille before. I heat treat the things that chainmaille is supposed to stop. I might be off about the temper temperature which is thankfully a fixable mistake, but do not take what I have said in totality I am just one of the simple voices of the internet (albeit one specialized in this). Do your own research too!