Make an Electron app that simply wraps your website! Just build in best-practices updating of the wrapper as well from day one, in case you want to ship improvements to the wrapper or start to move more things to client side processing.
As a side benefit, you get real estate in people’s docks and desktops :)
Would that help with the problem of integration though? What would be absolutely killer would be to emulate a USB HID keyboard or something, which would make it usable with pretty much everything, though there are definitely some security considerations there. Or if there are higher-level APIs to hook into that could work, but I would guess those would also require native function calls.
The way Google's keyboard works on Android, but on my Linux computer (and my Android phone) would be my dream here. I'd pay $10 a month for that for sure.
As a side benefit, you get real estate in people’s docks and desktops :)