Many co-ops prohibit sublets by shareholders who currently have mortgages on their units unless approval is obtained from the lender, and in some cases it does take time and effort to get the needed lender approval.
As the previous respondent advised, I'd recommend speaking again to your lender to see under what conditions an approval to sublet could be granted.
Your coop is protecting their investment and most banks do not allow this practice ( if they are aware about ) and in your case the building insists on this approval.
Typically you as a purchaser will get a better rate by purchasing as primary residence vs. investment. Speak to your bank again, they may adjust your terms, higher rate, some sort of a penalty. Explain your situation in full.
As for your building requirements, I adivse not to wonder and instead get a copy of your house rules and review them.
Good luck to you.