Yes... there was confusion with your comment "...treating for a potential ammonia spike..." and then mentioning some freshwater ammonia treatments. As you noted, those don't work in saltwater.
The stuff you mentioned now, Kent Marine Nitrate Sponge, deals with your nitrates... not ammonia. Two different things, but I'm pretty sure you know that. At this stage, I wouldn't be one bit concerned about your nitrates. In lower levels (<40ppm?) those aren't harmful to anything you have right now.
However... you've mentioned a couple times that your ammonia is "about" zero. Your fish are not going to like anything above zero. If you're getting any readings at all for Ammonia on your fish tank, you'd better be doing some small PWCs each day on that to keep the levels down until your bacterial filtration kicks in from the old/seeded filter media. Keep an eye on this aquarium the most. Your main tank will take care of itself as it is going to do whatever it wants at this point.
Again... I wouldn't rush it with the main tank at this point. Take your time to read up. As you originally noted, you started off with good intentions but made some bad decisions up front. That's fine, but slow down at this point and do that research you meant to do up front. As long as your fish are OK in their smaller tank, you have plenty of time. Regarding the big live rock tank, what's done is done.
I'm not sure why you're concerned about an ammonia spike in your main/live rock tank. If you see one, that just means you're going to have to re-cycle your tank. If your bacterial base is not sufficient to process the ammonia, you need to have ammonia present in the tank to build up that bacterial base. Getting rid of it at this point is not what you want to do - as long as I understand you correctly that you have no critters in there... just live (or not so live) rock. When most people cycle their tank, they don't do any water changes at all. I'm cycling my quarantine tank right now after I majorily disinfected it. I'm going on 4 weeks with the same water. It's gone from prisinte water, to 1.5 Ammonia/0 Nitrites/0 Nitrates, to 0 Ammonia/3.0 Nitrites/10 Nitrates, and now the nitirites are dropping. Eventually, I'll end up with 0 Ammonia/0 nitrates/ and around 40 Nitrates at which point I'll do multiple PWCs to bring the nitrates down to around 10ppm, before adding a fish That's the type of activity you'll normally see with a cycle.
I wouldn't say that your big live rock tank is "OK" just because you don't see an ammonia rise within a day. Personally, I'd probably throw some flake food (or the infamous dead raw cocktail shrimp that I'm sure you've read about here!) in there to give it an ammonia source, wait a week or so, and see how your water parameters test. If you see no ammonia OR nitrites, but see an increase in nitrates you're good to go.
By adding stuff (Nitrate sponge) to your live rock tank at this time, you're just making it hard on yourself to figure out what your tank is doing and where it's at in any cycling that it may be doing. Nitrates can easily be removed by water changes just before you add the fish back in. You don't need anything chemical to deal with it.
And yes... it takes an amazing amount of time to warm up 125 gallons of water!