Glacier National Park in mid/late September

Safety shouldn't be an issue in terms of snow travel, but might be due to hypothermia. In late September you might have excellent weather with cold nights, or you might have nasty rain/snow mix. I'd say it's worth rolling the dice.

Like Dave said, mid- to late-September in Glacier can be fantastic, or it can be rough–it all depends on when winter starts. I'd do it if I were you. It's one of my favorite times to get into the park.

In a lot of years, the first big winter-type storms will start in late September. These can be multi-day storms with a lot of precipitation.

This year, the first storm hit on September 18th and brought snow down to 5,000' and a lot of cold rain below that. We were lucky to finish our CDT hike that day. Other folks a couple of days behind us ended up having to road walk through the park.

If you're ahead of the storms, the great. If not, it would make for an unpleasant trip.

Yes. End of September that far north is rolling the dice for weather.
You could plan that trip and something further south with more dependable weather that time of year. Then decide between the two a few days before after looking at the weather forecast. Be flexible.

Billy