<<"Thanks. I was hoping that if we took the 5:30am train we'd miss the "rush hour" but maybe not!">>
Quite possible that you would.....
But, if you have the chance to be closer, you really reduce any uncertainties.
At least, the trains from Bern at that hour don't require a change at Zürich main station, although they pass through that busy hub.
I have done a similar thing from Luzern quite a few times, but since the trains from Zug northwards tend to run a few minutes late in the morning and especially during rush hour, I allow at least 30 minutes extra in addition to a comfortably early departure.
For travel to the USA, many airlines impose an extra level of security, such as an individual q&A session with a representative of the airline. Even though Zürich is a very efficient airport, it can be crowded at the early-mid morning departure time for most flights to the USA.
What time of year will you travel? In August, enough people are on vacation to reduce the morning crowds.
This link will help you with many towns and cities, including Bern as well as Zürich.
and, under where to go, each main city has a tab for "Hotels."
Those lists for the cities can include some extremely expensive deluxe places as well as less expensive ones.
My friends told me that the Hotel Walhalla, at Limmatstrasse 5, is another good choice in Zürich.
At the time they stayed there the hotel did not did not serve breakfast, but offered a coupon ( a "Gutschein") for a nearby Starbucks. And, they noted that it is very close to the "mid-platform" (under the tracks) pedestrian underpass, saving steps and time to reach the platforms. Also very basic.
I also have not stayed there. My friends are reliable, experienced travelers, by the way.
I'd look first at the Walhalla, before the Montana, based on their comments.