Submit an academic paper to arXiv built over a patented idea
To the best of my knowledge, arXiv has no rules against posting preprints that discuss patented material.  The only requirement is that you grant them a license in the copyright of the text you posted, so that they can distribute the article without having you sue them.  They have no such strict "open publishing" rules as you suggest.

However, in many jurisdictions, if you publish your invention (i.e. make it public in any way) before applying for the patent (or maybe before having it granted?), your patent can be invalidated.  Thus, you should consult with your patent attorney before publishing anything.