We are in the process of getting a developer ID authorized by our organization and hope to release a signed version soon. However we've been distributing the unsigned software for several years without issues. This is a scientific application so distributing through the App Store has not been a priority. A developer ID is not for App Store distribution. The App Store is a totally different thing. But if you want to distribute, or perhaps even run in the future, you're going to need a signature. Ideally, the app should be notarized.
We are replacing the Qt UI with an Electron app soon. Could you please explain more about what is problematic with the architecture? As far as I know, we aren't using anything that's platform specific. Hold up there! You are going in the wrong direction!
First of all, if your UI interface is not that complicated and you are considering switching from Qt to Electron, then you really have no excuse not to just do it in SwiftUI. It sounds like your UI and back-end code are well-separated. Otherwise such a switch wouldn't be possible. OK. Keep the separation. Just do the UI in SwiftUI. Do the back-end with a dylib or helper tool. It doesn't matter much. Once the app is hosted in Xcode, then these problems that you have now, and the massive problems you have coming up that you don't know about yet, simply go away.
Secondly, things like Electron, Qt, bash, and Python (and Xcode and SwiftUI) are most definitely platform specific. There is nothing wrong with that. The trick is, you have to use them on the appropriate platform. You are using, and switching to, tools that are not appropriate for the Mac platform. Can you get them to work? Maybe, but you are setting yourself up for so much pain - so much pain.
You said you have a scientific tool. Does that come in a command-line flavour? If so, you're essentially done. Use that. Wrap it in SwiftUI. Have Xcode build it all. Archive > Export > Upload > wait 2 minutes > Notarized!
(I realize I'm glossing over the bash and Python bits. If your command-line tool is Python, then that's still going to be a challenge to integrate into a Mac App. )