Hi James,
Great question! My two cents would be as follows:
Firstly, I'm assuming that there is no way that you can make the responsive pages show up with the same domain as the current non-responsive site.
I'm guessing that you are building these pages on a different platform than the main site, but it is possible to use multiple platforms on a single domain (for instance, many websites build their blog on Wordpress and their main site using something different) - so I'm curious about why in this case you are choosing to make it a totally separate domain.
Assuming that this is the only way to do it, my first question would be: are these pages something that would be indexed normally on the main site? You mention they are 'Action' pages - search engines may not need to index transaction pages, for example, where the only purpose is to submit a payment or sign up for something. Unless you also expect these pages to rank as entry pages, but if users are navigating to these pages primarily from a different landing page, they may not be relevant to search engines at all.
If this is the case, and you are still concerned about appearing to send users to a different website, you could consider making the donate button to these pages initially link to a page on the existing domain (like www.maindomain.org/secure) which could then immediately redirect to the secure version. You could nofollow this link and block the transaction pages on the other domain from search (I usually see no reason to have purely transactional pages indexed, but I may be missing some context in this case.) However, this may run the risk of looking like you are trying to do something sneaky with redirects.
You could also include a note on the first page (the one the user is on before being taken to the secure site), next to that 'donate'/'signup' button/link that states "by clicking this link, you will be taken to our secure platform at 'domain.com' to complete your donation/signup".
I do like the idea of including the name of the charity in this secondary domain if possible, to help make it clear that they are connected. You could optimize the pages' meta data to include the name of the charity as well.
I don't believe there is a negative consequence (generally speaking) to sending a user off-site - unless you appear to be running some sort of 'doorway' page scheme, or using sneaky redirects to show Google one thing and the user something else. My one caveat: I'm not sure what you mean by "booting" the user- I could see it potentially seeming problematic if the user is automatically sent somewhere without choosing to navigate to the other domain...it could even fall under the category of 'malicious behavior'. But as long as the user is choosing to click on a link that takes them to a 'donate' or 'sign up' page, even if it's hosted on a different domain, I don't see that creating a major issue for Google.