First you should change your setup so that each supply can be activated independently from the others. While testing each supply, look at the Ith pins and the actual inductor current to make sure they track each other. Also try adjusting the soft start settings on the supply. It sounds like you may be tripping the OVP repeatedly, which would definitely cause trouble, and increasing the soft start time would help prevent that (lowering your control loop bandwidth may help also).
Also make sure there aren't any shorts on the QFN packages. They can be very tricky...
I can activate the 3.3v and 5v separately from the 5v that powers the MCU. The only non-controllable output is the low power 5v output which powers the control logic. And when it is on by itself, it can run stable indefinitely. Now in testing I have the large delay and then "enable" signal going to the LTC3853 from the controller for just 1 output.
I'm not sure I have a way of measuring the inductor current unless I solder it up in the air and wrap a loop around the airwire. I have tried to look at the voltage difference across the output resistor, but my oscilliscope is not isolated so I need to do a math function on 2 inputs, and I am not sure how much I trust that.
I will try the softstart. Should just be a capacitor change since it is a low current charger.
The over-voltage protection on the LTC3853, or on my bench supply? I am pretty sure my supply can take it and I have tried 2 different kinds with similar results. It should be good for 30A at 12v. Looking at VIN it stays fine even when the inductor gets hot and squeals. I wasn't aware the 3853 had OVP, I thought it only had current protection.
I am not sure how to lower my control loop bandwidth with this chip, I figured all of that was internal and not adjustable. I have no external opamps or anything of the sort.
The QFN package was my first guess, but I have double checked under the microscope for connectivity and no shorts on the visible pins. The only thing I cannot see is the exposed pad. I did take it completely off using a heatgun, cleaned it all up and put it back on and the same behaviour. I also cleaned it in a sonic cleaner just to be sure as well.