If you sue the board and allege that the decision making was legally improper then the board would have to disclose the reason for the rejection.However, based on the actions the board took and the documents the board requested it is absolutely likely that the board will cite the concerns they had about your finances as the reason for the rejection and no other reason.
Generally speaking, the board would not need to prove that their concerns about your finances were justified, just that they had concerns, if at all. So they will likely present the same facts that led them to request you to deposit 4 years of maintenance as a valid reason for concern and the court will generally not dispute that. Even if you demonstrate to the court that you could have complied with their request to deposit 4 years worth of maintenance, the court would still likely not question the board's actions because legally they can reject you for any reason as long as it is not based on discriminatory reasons.
As noted, this board seems to be sophisticated and whether they had a legitimate concern or not, the paper trail would show that they had a concern about your financial ability which lead them to reject you.
You mentioned that this case goes against your common sense and I completely understand that, so let me explain that from a logical perspective a coop is a private association of people, a club if you will, and such people have the absolute reason, right and power to admit only those people whom they want and can reject others for whatever reason. From a legal perspective, a coop is a corporation and the owners are shareholders
. You as an outsider do not have a right to demand to buy shares and become a shareholder. Imagine if you did have that right then you could approach any private company and demand to become a shareholder in that company. Obviously this would not be acceptable. This is why even if it is found that the board somehow acted fraudulently, the court is more likely to award damages than to order the coop to sell shares to you (courts generally do not force people to be together if one party does not want to).
In conclusion, if you do file a law suit you would need to allege discrimination and/or some fraud for the law suit have some likelihood to proceed to trial as without it, under the business judgment rule that gives boards a very wide latitude in decision making over their private corporations
, your law suit stands a good chance of being rejected in summary judgment or earlier (meaning before it even goes to trial).
You are free to initiate a law suit and see where it takes you but I must note that the costs to you, even if you only go as far as summary judgment, would be fairly high.
Thank you again and please rate my service.