When you say an expensive tool is not an alternative, have you thought about testing both things separately?
To the test the JavaFX application, you can use frameworks such as TestFX. For the
WebView, you could come up with a browser that uses the same engine as your JavaFX application—which is WebKit—and test it there. You can determine the WebKit version programmatically:
The output will be a string similar to this:
Mozilla/5.0 (Macintosh; Intel Mac OS X) AppleWebKit/538.19 (KHTML, like Gecko) JavaFX/8.0 Safari/538.19
If you deploy your own runtime, it should be sufficient to use the corresponding WebKit version for testing. If not, you probably have to figure out what JRE versions (and, therefore, WebKit versions) your customers are using in order to set up an appropriate test environment.
This may not be super satisfying, but still better than no automated tests at all.