you can reboot the ONT via the Verizon Outage Page www.verizon.com/outage
on that page it says "Resetting my FiOS services once the power comes back on Reset your Optical Network Terminal (ONT)"
If you have a smart phone on 3g you can probably go in there and do it from 3 or 4g. Obviously your internet is down so you can't do it from your FiOS.
OR instead of the big reboot procedure you are doing, sometimes it's good enough to just hold down the Battery alarm silence, and the emergency battery use button on the Battery Back up unit. I think you hold them down for about 20 seconds, and that is supposed to reboot the ONT as well.
As to the latter, for some reason that doesn't work on my model. I know it's supposed to on all models installed since 2009, and mine was just installed this year, but holding both of them down does nothing :/
Something is wonky as this shouldn't happen. Might be ONT on the Fritz. Could be the BBU / PSU also going crazy.
Are you sure it's not a router issue?
I recently signed up and was using their latest "N" Actiontec Gigabit router and literally every 30 minutes it locked up and I would have to reboot it. So I went to local FiOS store and traded it in for an old Revison E. It's been a solid replacement until I can buy the real router I want.