It really all depends on die-off (for the most part).
if the rock you are getting from BRS is shipped dry but was just wet, then it will have some die-off (which leads to the ammonia). if you put enough live rock from the existing system to turn that ammonia---nitrite---nitrate fast enough then the ammonia will not be strong enough to hurt a fish.( the sand and water will also house the nitrifying bacteria) . this is assuming the eco-rox are cured at the facility.
If the rock has been and is now dry then there will be no die-off. and you will be okay to add a fish.
keep in mind it is all relative. you can only add what can be handled by your bio.
it all depends on what you get in the tank from the old and what condition the new is in.
if you use cured rock, a little live sand (and as much dry as you want), and say 1/3 total new tank water volume in the new, from the old. then you will probably not see a spike in ammonia at all.
if you tell me:
new tank size
how much LR from other tank your adding
" " LS " "
" " water " "
and condition of the Eco-rox I can give you a more accurate answer.
Wow, this turned into a long type-up. Just to put things in perspective, when I started my tank (long ago when everyone used hearty fish to cycle their tanks), I filled my tank with tap water ( horrid tap water at that), 20lbs/80lbs:live/dry sand, and about 20lbs cured LR. The third day it was up I put 5 chromis in. nobody died. Now this was long ago (when I did not know diddly-squat about the hobby), and I would not recommend this to anyone now. but, it just goes to show that a little bio goes a long way with some hearty fish.
All-in-all I think you will be fine with what you have described.