The only restaurant I have been to in Spiez is Seegarten where we ate pizza by the water. Looking at google maps the Belle Epoque is not near the lake and the photos show a not so impressive view - so the drawing card must be the food/atmosphere.
Speiz itself is actually not that amazing, there is no old town centre and the castle will be closed in March, also on a Sunday everything will be closed and you will find the place to be very quiet. I struggle to think of what you will do there to occupy your time other than walking around through the residential areas and a bit along the lake.
All of the tourism photos that I see of Spiez show areal views and they do look impressive. You, however, will be on the ground, and will not see things from this perspective.
There is a nice view when standing across the street from the train station and looking down at the castle and the Niederhorn mountain across the lake…but other than that, it is not really a location for great views.
Two alternate ideas I will float by you to research are:
For Bell Epoque stop at Kandersteg where you will find a selection of restaurants to choose from. There is a big Belle Epoque event there each February and hotel/restaurants to go with that theme.
Here you will at least be surrounded by beautiful mountains. If you have time and desire you can make a trip up to Oeschinensee - if the season is a good one there will be skiing there, but there are also groomed winter hiking paths or just simply walk in the Kandersteg area. Here is a nice one hour option
https://schweizmobil.ch/en/hiking-in-switzerland/route-441
Thun - much larger than Spiez, beautiful old town area (even when shops are closed), explore the two wooden bridges and water locks, walk up to the castle (doesn’t open until afternoons at that time of year) and enjoy the view.
https://www.interlaken.swiss/en/experiences/poi/thun-old-town
If you have a clear day the view of the mountains can be spectacular, especially from Schadau Park/ Schloss Schadau. You can see in one of the photos on their website what it looks like at that time of year. Lake Thun with Eiger, Mönch and Jungfrau as well as many other mountains in a wonderful panorama.
https://www.schloss-schadau.ch/en
I have not eaten at Schloss Schadau, but have had drinks on the patio to enjoy the view in nice weather and it is lovely. If interested you can always check it out. There is a nice little park to walk around in as well and it is within walking distance from the train station, or take the bus one stop.
Another possible location for a meal is Beau Rivage, along the Aare river (also a nice walking area). Friends of ours who live in Thun enjoy eating there for special occasions.
So, having been in Spiez multiple times at various seasons of the year - either hiking or more often cycling through to travel further along the lake - my opinion is that it isn’t really worth a special stop to visit. If the restaurant is a high priority for you, then for sure go for it, but I am not sure that the town itself will wow you.
