F-150 Ghostly Electrical Issues

I have a 2002 Ford F-150 with a 5.4l v8. The last few nights I have had strange
electrical problems. The 1st night, after a brief stop, I tried to start the truck and
the electricity went off as if someone pulled a battery cable. I popped the hood
and started to get back out and the electricity came back on without me having
to do anything. 2nd night, same scenario, but this time the electricity was off
long enough for me to check the cables. Very tight, no play, no corrosion, no
obvious problems. The power came back on within a minute. No evidence that
jiggling the cables was the cause.

Last year, my alternator went out and took my battery with it. I  replaced the
battery with an Interstate which was slightly larger than the regular size. The
original bracket would no longer fit. The tray is not much bigger than the battery
so I left it as is at the time since there was little room for play. After this 2nd
incident, I placed a bolt in the battery tray bracket spot to further retard
movement.

Third night, same scenario, same problem that quickly righted itself. But, this
time I started having issues while driving. The brights came on and went off by
themselves, several dashboard lights flashed on and off, the fauges stooped
working then came back on. These issues never occur during the day, and
acceleration exacerbates them. I replaced my alternator last year with an
inexpensive repaired unit, so I suspected it. i had it tested at AutoZone and it,
the battery and starter all passed inspection.

What cou ld be the problem?