Just dealt with this as well, but instead of mailchimp, I was writing into pardot. I did the approach a bit differently, but same general idea as Anton(taking the data and sending to 3rd party after submission). I cloned the original bizform webpart and created an onaftersave event which sent a post directly to the other service. This allowed me to reuse the same webpart and build in properties like url to post to, field mapping, etc. Worked well for me, but Anton's suggestion would most likely work as well, just may not allow you to have as much control, which in your case, may not matter as much, since it is just the checkboxes on a couple forms.
Let me know if you need more specifics and I can see if I can help.