I think caramelization might be the wrong word there. I would go for concentration. While some caramelization does occur around the rim of a pot of tomato sauce as the sauce reduces (and it's good to stir this residue into your sauce), the main increase in sweetness is due to a concentration of all things tomato-y as the water in the sauce evaporates.
Speeding up this process is not only possible, but, in my opinion, highly desirable. A long- simmered tomato sauce can often have an overcooked taste to it. One method to speed things up is to strain your canned tomatoes through a fine-mesh strainer, or something equivalent. Use what remains to start your sauce in the normal way and keep it at a simmer. The liquid, however, you can heat more rapidly at a low boil since there are very little tomato solids left in it to get that overcooked flavor. Once your liquid has been reduced by 60-75%, you can add this back to your simmering tomatoes.
If you start your sauce with fresh tomatoes, the quickest way to accomplish this is to freeze them. Once thawed, all their liquid will pour off easily. Boil down this liquid as described, quickly sauté your remaining solids, and then blend them into your reduced liquid with an immersion blender. This results in a very rich and flavorful thickened sauce that still has the brightness of fresh tomatoes and can be done in a fraction of the time mentioned in your question.
Tomatoes aren't high acid, so they need the addition of vinegar or lemon juice in order to safely can with a hot water bath. Honestly, I'm not sure why you would want to make tomato sauce from canned tomatoes because for me, the whole point of canning tomatoes is because the tomatoes will otherwise go bad. But anyhow...
I recommend finding a tomato sauce recipe from a canning cookbook or recipe website. Here is a recipe I found from the Ball website which is a reputable source, and here is an article about hot water bath canning tomato sauce. Notice that the recipe calls for a high acid ingredient, which is extremely important in canning tomato sauce with a hot water bath!
Secondly, it is NOT okay to seal jars by inverting them!!!! Get yourself a book about canning, Ball sells several good ones (Blue Book and Complete Guide to Home Preserving) and learn about the canning process. The steps are as follows:
(0) Follow the recipe EXACTLY, don't add any other ingredients that may change the pH of the final product!
(1) Get the hot water boiling in a jar large enough to have a few inches both above and below the jar that you'll use. You can use a metal trivet or canning rings to act as a base for the jars so that they have water going under them and aren't touching the base of the pot directly.
(2) Start with jars that are appropriate to canning and have fresh lids (or clean, working reusable lids like Weck jars or Tattler lids). If processing for under 15 minutes the jars need to be sterile (can be made sterile in a hot water bath). Inspect Tattler seals for any nicks, and throw away the bad ones.
(3) Normal canning lids and Tattler lids need to be prepared by soaking in hot water right up until use.
(4) Add the ingredients to your jars, remove air bubbles with a spatula, wipe the rims clean, and apply the lid as per the instructions of your lid. (Regular Ball lids are slightly different from Tattler lids, and I've never used Weck jars which I think are different again.)
(5) Insert into the hot water bath with canning tongs, cover, make sure that you start the countdown once the water has returned to a full boil.
(6) Remove once the time is up. Don't tilt the jars when removing them as the seal isn't airtight yet. Check the instructions for your lids, at this point Tattler lids have to be tightened fully.
(7) Leave the jars alone to cool. After a few hours or overnight, check the seals by trying to gently pry open the can. They should be "finger tight" and not pop off. The jars that aren't properly sealed can be put into the fridge and consumed within a short time. Otherwise they're good for up to a year or whatever the recipe suggests.
Best Answer
Anyone here who gives you an estimate beyond what a typical (non-canned) refrigerated sauce would last is just going to be speculating.
The thing about canning recipes from reputable sources is that they are tested scientifically. They often run hundreds of trials with testers for a particular recipe, then test each for bacterial growth, etc., before deciding a recipe is safe.
And the somewhat unfortunate thing (for your purpose) is that most of that testing has been done with the goal of having an unrefrigerated shelf-stable product. There are a lot fewer recipes and processes that have been tested that way for products that will be sealed and refrigerated afterward.
That said, the growth of the market in refrigerated convenience foods in recent years has led to new commercial processes. At a grocery store, you can sometimes find things like refrigerated soups or sauces that are sealed and have a somewhat extended lifespan (from a couple weeks to a few months). They won't last as long as a room-temperature canned product, but there are additional processing steps taken to give them a longer shelf life. (Some of it is due to preservatives too, but there are often processing steps similar to canning to add to shelf life.)
To my knowledge, most of these processes are still primarily handled on an individual basis by commercial producers, who often have to do testing themselves to ensure their products will be safe. I don't know of any reputable resource that discusses home recipes for sealed jars under refrigeration and their shelf-life. (The other aspect to the traditional canning recipes is that they are usually structured to keep stuff safe indefinitely under storage, rather than targeting a specific shelf life.)
So, in your case with an untested recipe and untested process, it's all just speculation. The biggest factors in spoiling of tomato sauces in the fridge are often due to mold and other spores that get into jars, so it's probable that your technique will stave off that concern a bit longer. But as to precisely how long your sauce will be safe, no one here can tell. Without testing, there's no guarantee it will have a safe shelf life beyond that of any unsealed refrigerated sauce you put in Tupperware or whatever other container.