A few other ideas:
I would vinegar wash/scrub your mixing vats since you have phosphate testing there and are using RODI water. If you're not using containers that leach or retain phosphates (even some rubbermaids do), then you still could be fighting yourself if you used them to store/cook/clean any live rock/sand. If your mixing vat has phosphates showing in tests, I think you need to resolve that issue.
-You could get your water tested somewhere to make sure you really have ammonia. I've heard the API tests sometimes register false positive on low ammonia (or that people are just using the color chart wrong). Also, I suggest you use the Seachem phosphates test kit, not the API. The API is not nearly precise enough (for me, API mine showed around 0 when in fact Seacheam showed I had .06, obviously very bad).
-Bacterial supplements have really helped me with some of the startup short-term ammonia issues. I had rock that was fully cycled at my house, but after 3 months I still saw minor ammonia spikes when I was doing things to the tank (moving rock, adding fish, etc). You can't really overdose on them if you're having all of these ammonia issues. It is better than Prime in some situations, and Prime can interact with other products even though the label doesn't say.
-IMO the newer rock may be the issue, like ReefChief said. It probably needed to go through a full curing cycle since it wasn't bleached etc but left to dry for 24 hours, and you could just have slow sponge deaths and ongoing die off that continues for awhile. You could up the rocks and pull off any more die off with tweezers, keep doing heavy/frequent water changes to dilute the ammonia spike, and do the bacteria supplements.