Definitely take your bags with you. I would go straight to the restaurant car as you board. If you are using the SBB app, tap on “show train information” and it will tell you the composition of the train. It will also be on the sign that hangs above the platform that your train arrives on in Bern.
I was on an early morning Bern to Zurich train this week and sat in the restaurant area. The seats were larger than they are in the regular second class cars and I was able to put my large backpack under the table at my feet. It sounds like your luggage is fairly compact, I think you would have space for it.
Also, I asked someone I know who takes this route for work and they were pretty sure you would find luggage storage just outside the dining area, as you enter the train car. Take a look there as you enter.
The trip is one hour, a little longer if you are going past the Zurich main station to the airport. Plenty of time to eat breakfast. There are many direct trains. To be sure there is a restaurant, just look for the knife and fork restaurant symbol when you make your choice of which train to take.