Does your colleague (another brand manager) have the authority to issue such a letter? You are a bank manager, you should be able to answer this question on your own, can you issue such a letter to someone?
Anyway, a lot of private companies do trouble their employees while giving our reference letters, in such case, you can submit copies of your payslips, your offer letter, copy of ID card as proof of employment, if you were promoted to a Bank manager level, you must have been given a promotion letter which must have explained your responsibilities and pay and perks, that letter can also be used as proof of employment and experience and certifying your responsibilities as a bank manager.