How does the trigger behave when you pull it without the upper installed? Is there any binding? Did the spring set feel like it lightened the trigger? If you set the new spring next to the old one, is the deflection angle increased slightly (i.e. a lighter spring set needs at least the same, if not more, preload to reset properly).

If all you have to do is reset the safety on and off, then it will fire, then the hammer is obviously engaging the disconnector, but it must not be re-engaging the sear. That's where I'd be looking. One reason for that might be that the trigger isn't resetting completely forward, so either the spring isn't indexed properly, or it's binding, dragging, etc. OR, the trigger return spring isn't strong enough to reset on its own. You might be able to increase the pre-load on the spring by either bending it or blocking it. Might lube the pins, or even give them a shot with some polishing compound. I rub polishing/lapping compound on pins, then spin them in the trigger to help polish the inside of the pin hole as well, then spray it out with lube, to help ensure the trigger is 100% free. Not sure this could happen with an RRA or not, but it also might be that the disconnector is binding in the trigger so it can't release the hammer to engage on the trigger, so when the trigger resets, it's not releasing to mate the sear.

I'm not as familiar with the RRA 2 stage as I am with some of the other triggers out there, but that's where I would start. Play with the lower without the upper installed. Can't just put it together, have it not go bang, and not dig into what the root cause would be.