The bank is always free to waive a fee. The integral issue here would be ensuring that the applicant understand the dates and that the bank must have the application on or before X date to be eligible. I trust the bank will have a way to determine the dates the pending applications were received in order to enforce the cutoff date, or to review the postmarked date if that is used. The latter will assist the bank in defending itself from the upset customer who swears he mailed in his app in time and the bank just held it to avoid the free offer. Enough claims like that and you have UDAP claims.
If this special is only advertised in a branch, is the branch located such, and serving a diverse selection of the market such that the bank is insulated from claims of discriminatory lending? (Example; Branch A is in a low-income area, Branch B is in an upper income area. Only Branch B will offer the special. That would be an issue.)
To avoid criticisms, the bank may opt to put stickers on the applications emphasizing the postmark date requirement and removing them after that date. Some form of a virtual sticker/notice may be used if this is offered online as well. Just to ensure there is a valid business reason for the limited offer, I would recommend the bank be prepared to define it's position if this is offered in limited areas such as Branch A or Branch B only, but not both, or not in other products offered online.