With that said.
1. You can do a water changes, at least to get ammonia down to 2ppm. With fish-in cycling, the fish are constantly producing ammonia anyways, so even doing a water change to get ammonia down to 0 won't affect the cycle, not really.
2. When you say you started with live rock, was this like, actual live rock immersed in saltwater at the store that likely already has beneficial microbes already established inside? Or are they dry rock, or perhaps just random rocks the LFS chucked into a saltwater container and call them 'live rock'? If you establish a tank with all live rock (presuming you have enough as well, of course), then there should be plenty of nitrifiers to take care of ammonia and most likely allows for insta-cycling. It's honestly uncommon for 'good quality' (sorry, I don't have a better way to describe it atm) live rock to not be able to oxidize heaps of ammonia, especially after a month of being established.
3. That brings up this question - how much live rock, sand, and other biomedia do you have? Sorry, I can't really tell from the pictures, I am super bad at estimating measurements.
4. I would never recommend cycling with so much live stock in, it is just so much trouble, as you are going through right now. And costs too, you are buying ammo lock, testing constantly, probably gonna painfully do a bunch of water changes if need be, etc. Just so much easier to do fishless cycling, imo. But anyways, that's my preference so I guess I digress.