If the rules of the coop are that a board package has to be submitted in order to rent your unit, there is no getting around it. You say that real estate agents are apprehensive of going around the board and not submitting one. The fact is, though, that they have no choice. An agent cannot rent out the condo and then have that renter be at risk of being put out by the board because procedures were not followed. Condos have what is called the right of first refusal. On a sale that means if they don't let you sell it, they have to buy it. For a rental, if they don't let you rent it, they have to rent it. No board wants to do that so they do have to issue a waiver of the right of first refusal, and in order to receive that, the proper application and back-up have to be submitted.
If they exercise their right then they have to place or find a tenant for you at the same rent and terms of a "bona fide" offer you submit to them in the board package or they have to waive that right and let you lease to your applicant.
For many reasons many Manhattan condos have "coop envy" these days and are slowly moving toward the co-op board model but they don't have the same power and control like a coop board.
Lic. Assoc. R.E. Broker
The Corcoran Group
You say Condo. If your bylaws require the application, that is that. If they're hostile towards you, it makes me wonder why? Good luck.