I can attest to the FCC working with you. At one station the CP expired on an upgrade. My boss had our attorney walk over to the FCC to ask for renewal stating my boss had always taken care of his business in the past on a timely basis but in this instance he was the caretaker for his spouse during her illness and buried her only a few days ago. He got an instant extension. And the attorney said he would ask but my boss was wasting his money by having him ask, as the attorney said an expired CP was never reinstated.
Now, as I understand it, you are licensed at one spot and have an active CP for another site. You wonder about shutting down the site with the active license and building out the new site prior to the CP expiring (I'm guessing the CP is a fairly recent thing). You are wondering if you shut down, go silent and stay that way until the new site is ready to roll is okay. You are saying you will not have the new site ready to go until near the date the CP expires. Hopefully I have that right. I wish I could give you the answer.
That really begs the question as to why. The only logical reason is financial. An example of this is not always a 'no money' issue but throwing money at a bad investment when the money is better spent on the upgrade. That can be a legitimate reason. Say you have a LPFM that looked good on paper but is a nightmare in reality. To continue to keep the existing station on means it is difficult to find the dollars to construct at the new site. Obviously the ground work and PR you have done with the existing station is worth something but perhaps the station continuing to operate is making it a financial liability concerning the new build out. If the station is holding its own, why not stay on the air? If it is costing you money it would be better to correct the negative trend as fast as possible getting on the new site as fast as you can so you can start reversing directions financially, not waiting until near the end of the CP if you can help it.
It could be an issue of a leased tower site with a CP for your own site. For example, your lease renews and for a term you don't want. If you don't renew, then you lost your lease.
I'm grasping at straws obviously, but realistically I cannot see any reason to throw away what you have established thus far in order to go silent until you're at the end of the CP unless you would be losing money or that those dollars are needed for the new build out.
I guess the big question is why do you want to do this? If it is the best scenario, then a heart to heart with the LPFM branch would be a good idea. If they know your situation they can help and they might just tell you how to make things work in your favor. They can't give legal advice, but the truth is they typically don't talk to a bunch of licensees so by merely talking to them sort of makes you special, no longer just a faceless name on an application or license where an engineer speaks on your behalf. The FCC isn't out to get you but rather to try to help licensees do what they need to do and get it done the right way.