What are my chances of actually using the 125 watt 6400+?
I'd say your chances are passable....before you ask the processor to do any real work. Google the processor and "recommended PSU" to get some internet knowledge about the lowest PSU someone has tried to run it on. Consider also what amount of amps it will want.
What will happen if I just try it and it exceeds what my PSU can supply?
Well for one, it may end up starving other components of your computer when the processor goes under load and your PSU capacity is met. I'm guessing your video card would be the most likely component to starve, but I'm just talking out my ass.
You might notice your CPU fan not spinning up to max which will cause it overheat, your system might struggle to in various CPU intensive situations, applications might fail.
Or depending on its idle power wants, your system may just go "FUCK YOU" when you try to boot.
Ideally you want more power than all your components ask for. You don't need a lot more, but you want more. Frankly, I've been using more than a 400w power supply for probably close to a decade, it amazes me anyone that has an interest in PC gaming uses less than a 350w.
All I can really say is, never underestimate how power hungry new hardware is, especially if you haven't upgraded in years or are making a huge jump in performance.
Instead of guess what your current power use is, too, you should go get a good diagnostic monitoring tool that will tell you exactly how much power your using, and how much you have to spare on that PSU.