For a single store, you will want a custom App. You do not need one that gets reviewed by Shopify and installed in more than one store. If you do, just select a normal App, remembering that it will be unlisted, but installed in more than one store.
Embedded or not makes no difference. That is just how it looks and acts within the construct of a logged-in merchant, so do not worry about that. I have no idea what your existing App in dot net does, but you are correct, you can hook up your Shopify App to your DB and use it as you wish in your App. You can point N Apps at a single database you know. I often build Apps that have a direct embedded install in Shopify, doing all the oAuth and merchant API key work, but the guts of the complicated business logic take place in a separate App where we hide all the connections to external APIs and processing. This pattern is fine. Shopify only ever read/writes a Shop table, while the business App works with everything else. Think of drop shipping Apps and bookings, and other complex stuff. No need to encase all that in a monolithic Shopify App, when you can indeed keep that aspect of things dead simple.