Quality content, I couldn't have said it any better myself!!! You hit every point.
Just to add: Here's some Binghamton specific resources on the topic.
SUNY Patent Policy & Royalty Distribution
As you predicted, the use of resources is the ultimate decider. If OP's business does generate meaningful returns or gains signfigant exposure, the University will very likely use any argument try to seize ownership for their own gain.
Like you said, if this project is going to be anything more than a resume boosting side project, don't utilize any University resources.