Absolutely. White bread spikes your blood sugar much faster because it's almost pure starch. The resulting drop can definitely be sleep inducing in my experience.
Whole grain bread also has a lot more protein, which is also more filling. But can also be more sleep inducing. The topic of post-prandial somnolence is a complicated one indeed. All sorts of hormonal, blood sugar, psychoactives in some foods, factors involved. And still poorly understood in many ways.
Sort of related: if I eat any amount of white bread or other processed grains I tend to get a lot of abdominal pain. I have a bunch of scar tissue down there due to appendicitis with bad complications many years ago. Whole grains help keep me pain free, it's the only consistent remedy I've found.
Do you live outside Norway? Is Jarlsberg expensive as an imported product? It's pretty affordable here so I might be amiss in listing it as the cheap option.