You can also experiment with CEF, using its Swing component. See here to learn more about that and how to deploy:
https://hydraulic.software/blog/13-deploying-apps-with-jcef.html
The primary limitation to be aware of when doing stuff like that is z-ordering. Overlapping stuff with the browser won't work and that includes things like context menus, dropdowns etc. So you have to design your UI with it in mind.