Thanks for clarifying.
I don't like the CON check idea, because your CON bonus is already reflected in the additional hit points you get from it.
BUT, I like the idea of the variable DC for first aid Heal checks. That makes a lot of sense. It would make it more urgent to get to a dying character, rather than letting him bleed for a few rounds while you finish off the foes before you administer first aid. I don't think it would create any major balance problems since the average DC (ranging from 11 to 19) would be 15. It might make dying slightly more frequent though. Here's why:
Let's say your buddy is down at -1. He is unlikely to end up dead. (DC 11 is easier than DC 15). On the other hand, even using the standard DC 15 the PC would be unlikely to die because if you fail at your first attempt, you could always try again. You would have several rounds to re-try before he actually dies. So, really, your modification does not make it that much less likely that a character would die.
Now let's take a different for-instance. Let's say your buddy went from 5 HP to -9 HP from a single hit. You have to heal him in one round or he'll be dead. There is no opportunity to re-try. So, in this instance, the character is much more likely to end up dead if you're using the variant rule that you propose.
You'll end up with more dead characters.