I think this thread will help, even though its just focused on the theme of the iframe created by the google tool:
https://forums.modx.com/thread/99538/recaptcha-v2-how-to-change-it-039-s-theme
basically in recaptcha extra there is a snippet which does the work, independent of your form.
I don't know your level (I am not much of a coder unfortunately), and the proper solution probably is to figure out why it won't validate. I tend to work around problems due to my lack of ability. There is always another way to skin a cat, and sometimes that way is totally inefficient but still works ok.
Anyway, my thinking is this: treat the iframe, which is totally independent of the post itself (the form as you say) as an overlay on top of the "post" button. That way its two steps, first validate, then post.
Hmmm, not sure if this is the right way...the more I think about it the trickier it gets. And you say the hidden field also won't work?