NavList:

Definitely a fun puzzle, but why do it with a time sight? Not that you can't, but for most folks with cel nav skills, it's probably easier to do it by more common, familiar sight reduction methods: calculate H_c at two different times. This is actually what Letcher recommended. Calculate H_c for, let's say, 5 minutes before the estimated GMT and also 5 minutes after. Then interpolate with your observed, cleared altitude. That gives you your time correction. You could easily do this with USNO's online celestial data page. And, by the way, I got the same result, more or less, that Greg Rudzinski posted using this simple technique.

But with the data you've given, there's something much more interesting that you could do, potentially. You could ignore your known longitude and use the Moon's altitude from the sights on the previous day to determine GMT and then get local time from the Sun altitude the next day. Now you've got some navigation fun! It's not terribly accurate, but altitudes of the Moon are a poor man's lunars.

-FER