So I set up the game on my mac at work blush and tested this.
If you beg gold from someone you already have a peace treaty with, you form a 2nd peace treaty that will be canceled 10 turns after the beg.
The 2 peace treaties will end exactly as if you had just one of them. So if we beg gold on t110 we are safe from a DoW from Hammuragawa until t120 regardless of when our 1st peace treaty expires.
In my test I declared war on an AI, made peace, liberated a bunch of cities to get pleased again, then begged 5 gold from them. I had 2 peace treaties with them. I ended turns until the first peace treaty expired and the 2nd one stuck around.
I know for a fact that this happened to me. I'll try to test and figure out what conditions allowed the AI to cancel peace a couple turns after begging for gold.
Maybe they can cancel the deal, but just didn't in your test. I'm glad to hear Kaitzilla is going to try not to overlap the peaces anyway just to be safe.