I was able to concentrate the problem on two possible solutions:
1. Editing some javascript/HTML code to prevent page from redirecting after "submit" was called. I've been having problems to get this properly working with jQuery and I'm not sure how I should put it together using AJAX. I'll have to look into this deeper. Apparently there's also a possible solution using iframes, but it seemed quite confusing so I haven't tried this yet.
2. Disabling captcha is not an option. But if I can force one specific captcha to show up every time instead of being generated anew, then the problem is solved. Sadly so far I haven't been able to do so. ._. Currently I am going through
Google's Recaptcha source code to find out how it is determined which "challenge" is currently used. Apparently the challenge is an encrypted string that is supposed to identify the current image/text. Or it could be just the encryption key. This would be confusing, though, as the private key is used for that. In other words, I actually have no idea yet what is what. I'll have to continue analyzing this tomorrow.
The limitation with this approach is that I have only the final HTML source available that I can modify as much as I want.
EDIT: Turns out that the "challenge" for the captcha really is an encrypted or hashed form of the solution as it differs every time. The downside is that it's protected against repeater attacks so a captcha generated once can only be "solved" once. Sadly this renders any other way to attempt a repeater attack pointless as I would need an automated captcha solver. ._.
I guess I will have to continue my experiment by manually doing that big chunk of work that I wanted to automate. ._.