@Adam_Burge, I’ve reproduced the issue you describe now on my Mac, also running Catalina. The keyboard works as expected in TextEdit but in Google Docs in Chrome Shift+X, Spacebar produces just
ẖ, and does not transform to
𓄡 as expected. This behaviour is the same in Safari.
(Interestingly, in this forum post, also running in Chrome, it seems to work just fine!)
In Google Docs, it appears that if you do not release Shift before pressing Spacebar (i.e. press and hold Shift, press X, Spacebar, release Shift), then the transform works as expected. This is helpful diagnostic information for me and may be an acceptable workaround for you? I think all the combinations in the keyboard allow for this (as the sequences that require capitals are always single letters as far as I can recall).
I don’t have Adobe InDesign installed here on my Mac at present but can install and test at some point. If this workaround gets the job done for you, please let me know!
I have created a bug report about this problem and we will be looking into it further.