No, because a small business that does a lot of cash transactions would, on review of the account, look like a business, not like a money transmitter. There would be checks written to vendors, withdraws from a payroll company corresponding to a predictable cycle, probably a rent check on a predictable cycle, etc.

If your only two transactions are "Deposit lots of cash" and "transfer it to another bank" the bank will come to the conclusion "To the extent that there is a business here, it is probably money laundering." The bank was right.