There are about 12 fundamental constants of the universe that, were they diferent even by a TINY fraction of their current value, would lead to a universe fundamentally different from the one we inhabit.
This is a red herring. Cyberia, there is a constant called "pi". Its value is extremely precise and if it were different, the universe would be unimaginably different. But, there's no evidence that it could possibly BE different. The ratio of the circumference of a circle to its diameter is what it is. So a discussion like that is putting the cart before the horse. There is no evidence whatsoever that the so-called fundamental constants even could be different, any more than the ration of the circumference of a circle and its diameter could ever be anything other than our current known value for pi.
The real issue here is the term "constant". If you can change it, then it's... NOT a constant. It's a "variable". Variables are different than constants. Naturally, if you arbitrarily decide to start changing constants into variables, things are going to start being different. Is that insightful? Is that clever? Does that mean anything? I don't think so. That's like taking a cat, pointing out how alive it is, then holding it under water for ten minutes until it drowns, then taking it out and pointing to how dead it is, and saying, "Look how different it is." Well, DUH!!! You mean when you change something, things change? If that's the best argument for a designed universe, I'm fine with that, because that's a shithouse argument.
It is assumed (probably correctly), that most, if not all, of other configurations of these constants would NOT allow life as we know it to form.
What do you mean "as we know it"? Again, things would change if we changed things? Things would be different if things were different? Next you'll be telling me that when I chop my hand off with a big knife, I'll be in two pieces.
You know, when I go to the store, I like to buy a plastic tub of a product named "Mixed Nuts". I take my mixed nuts home and I eat them. I can eat a handful of nuts from the tub and what is left is still mixed nuts. I can keep doing that and keep doing that, but at some point at the very bottom, there may only be peanuts left. Even though that last handful of mixed nuts I removed looked the same as the first one, now, something has changed, and the container no longer contains "Mixed Nuts". Now, there are only peanuts. Eventually, there may only be one peanut on the bottom. That is not "mixed nuts". That's a peanut. Even though there is product left, it's not the same product as there was before. A peanut by itself can never be "mixed nuts". Is that an amazing observation of some kind? Is that a deep, earthmoving philosophical moment? No. It's just the thing that happens when things change. Something changes, things become different. It's not really very amazing at all.