Regarding classic confinement, there is the Process for reviewing classic confinement snaps which will give you some idea of what that looks like - basically to be granted this, a snap must both require it in order to function (ie it is not possible to get it working under strict confinement at all) and the snap must fit within one of the existing supported categories for classic confinement. However, I think unforunately this snap may not meet the second criteria since I don’t think requiring access to a particular hardware device is not a supported category.
Also if your snap is in the public store, whilst it may not be intended for use by others, they could still install it.
As I suggested above, I really do feel like this sounds like a great use case for Ubuntu Core rather than classic Ubuntu (or some other Linux) since then you can prepare a custom image for the device which contains your own custom gadget snap which in-turn defines an appropriate serial-port slot to provide access to /dev/ttyS0 for your snap. See Building a gadget snap | Ubuntu which mentions the 64-bit PC Gadget Snap which sounds like it would be a good basis to work from in your case.
Beyond that, perhaps @ogra may be able to provide further advice if needed 