Switching over a 600L from SB to BB

[Almogia]

Title says it all :P

(NOTE: The aquarium pictured is not the one I am having trouble with. The image is just to illustrate the set up)

My aquariums house frags for a research lab (studying the effects of climate change), so, unlike a display tank, my corals are all resting on egg crate (and aren't connected to rock)

[When the aquarium was first set up (about six years ago), the idea was to create a system that is as similar to the ocean as possible. However I have realized that this is an extremely impractical approach.]

This specific tank houses Leptopsammia and Balanophyllia. Temp is at 17 C, lighting is metal halide, there is a protein skimmer and an alkalinity reactor, nitrites/nitrates are at zero, calcium and magnesium are a bit low but stable and I have started supplementing, and the other day I checked phosphates and they were THROUGH THE ROOF.

I realized that the urchins in the tank aren't doing a great job at cleaning up the huge amount of food I put in there (Lepto don't have zooxanthellae, so they need multiple feedings every day).
Additionally, as I am relatively new to this, I didn't know that I need to, and thus never have vacuumed/siphoned the sand (I'm still not even really sure how that is done), and I highly doubt that the guy who was working this job before me did either.

So recap:
- huge amounts of food causing phosphate buildup
- cold water means I can't toss in snails to help
- five or six years of phosphate buildup means (if I understood my research correctly) that it's possible that the sand has actually absorbed a lot of it, and will release it if I leave it in there, so sand can't just be cleaned.

My conclusion was that the sandbed must be removed. If I go BB and cook all the rocks that are in there, it'll be easier to clean up food and waste detritus manually, and the rocks will retain their bacteria so I'll still have them to carry out the nitrate cycle.

And now, we reach my question:
Is there a way to carry out the process of switching out SB to BB, and cooking the aquarium rocks, gradually? Or must I do it all at once, pull out the corals and suck up all the sand in one fell swoop?
I understand that leaving dirty rocks in isn't going to help the water phosphate levels, but if I pull out both rocks and sand, then I am pulling out all of my bacteria, and what will carry out the nitrate cycle until the rocks cook completely (which takes at least a month, I think)??