The solution is to download the ZIP file attached to this post (see below) and overwrite your two existing files here:
- /plugins/system/relib.php
- /plugins/system/rsfprecaptcha.php
Once you get to this step, refreshing the form should give you an "Invalid API Key" error instead of the old V1 system.
Last, all you need to do is sign into the admin panel and go to Components -> RSForm Pro -> Configuration and enter your new "V2" API keys on the ReCaptcha tab.
Done!