Your requirements are mutually exclusive. I don't know of any way to get the combination you are describing.
If you are really looking for "airy inside", then you need to get as much rising out of the bread as possible. The air isn't present in bread dough before baking, the airiness of bread happens by the yeast creating and inflating myriads of tiny "balloons" whose walls get set by the oven heat. Just the way you can't inflate a balloon without it getting bigger, you can't get airy bread without it getting bigger. There are breads which rise relatively little, but they also don't have an airy interior, e.g. pumpernickel.
If you would drop your "has to not rise" requirement, then the airiness is easy, it basically happens automatically with any proper recipe. The crisper crust is a bit more involved, but in a sweet bread, it's not that difficult, just look for recipes with enough milk and sugar in them. Also, you can look at older questions about crust around here. Beware that a mold and a crispy crust also don't go well together, a crispy crust needs free exposure to air.
The ultimate combination of very airy sweet bread dough on the inside and a very crispy crust on the outside is a donut - which means that you have to leave all molds and even your oven behind and start deep-frying perfectly expanded spheres.
If you instead insist on having the bread fit a mold perfectly - e.g. if you have a certain complicated shape and want to bake it in a silicone mold, - then such pastries are usually not made with bread dough. If you need something that's relatively close to two-dimensional (some relief is OK), then there are different doughs close to cookie dough which can make such stuff. Gingerbread is one option, but you can also look at other classic shaped pastries, maybe also look into recipes for easter lambs.
If it has to be thicker than 3-4 cm, then you'll probably need to make it a cake instead. There are cake doughs which perform relatively well when baked in a mold, I think that classic quark-oil dough is good for this kind of application. But also some well-structured batters take well to this kind of shaping, and are relatively airy after baking. You'll have to look around for such cakes, many will be labelled simply "cake" or "sponge" but in the end, some types of sponge hold up to such treatment and some don't, so try finding recipes which are meant for mold shaping.