The lime soil is creating a problem for your peach tree. It's possible that with the 2 main trunks, that the roots of the healthy side have gone in one direction where there is adequate nutrients and a more neutral pH, while the pale side has hit a more concentrated supply of lime.
It's a bit of a piggyback problem in that lime soil itself has a negative effect on phosphorous and potassium uptake, and if the potassium (potash) level is low, then the potash depletion won't respond to any nitrogen supplements (to green the leaves). In a nutshell, one part is dependent on the other. Until you get the potash part sorted, dumping on fertilizers and such won't work.
You'll need to first up the potash intake which can be with a nice layering of older wood wash all around the tree, particularly the depleted half. Or you can use cow manure which is specifically higher in potash than the others (2-0-3). You can apply the woodash and then the rotted cow manure on top and either water it in or wait for rain. Eventually you'll see see more life in the leaves and by next year it should start fruiting again. If any flowers appear this year, pick them off so they don't use up precious food supplies while the weak side recovers.
If you're in a great hurry, you can usually buy straight potash fertilizer.
I'd highly recommend that every fall you give the whole tree a nice thick mulch of rotted cow manure and you shouldn't see this problem again. You'll also get very sweet peaches.