Yes, safe, and also way safer than doing a visual approach without any backup at night. Like the other commenter said, the local altimeter setting is only required for knowing when you arrive at Decision Altitude, which is not applicable for this visual approach.

I'm not sure of ATC's rules, but my assumption is that they would NOT clear me for an ILS approach in this circumstance. But that's moot, I would not ask for it because it's not authorized anyway. It would be a visual approach, we'd just happen to fly the ILS course and use the GS for vertical guidance.

It would be safe. The ILS projects a localizer signal (left/right deviation compared to runway alignment) and glideslope signal (up/down deviation versus the intended flight path, usually about 3°).

The only thing you need a precision altimeter setting for is to determine the safe decision altitude (point at which you need to see the runway environment to continue, else initiate a missed approach) and a fairly theoretical chance to intercept a false glideslope lobe.

With the clear weather, knowing the decision altitude with precision was not important and the false glideslope is easy to identify by comparing outside view and descent rate needed to stay on the glideslope.

ILS, yes (the G/S vector and angle is calibrated to account for terrain), LOC on the other hand (i.e. non-precision with no G/S information) would have probably ended up in the same situation as it requires pilots to manually manage rate of descent. Not all runways have ILS equipment on both sides of the runway, let alone on all runways available or one that is active. At some airports it's possible you only run a 30% chance or lower that you can actually use the ILS to land, because the one with the equipment may not be the active one (and which runway is active changes depending upon wind and weather)