As noted, the law does not require a "foreign" corporation to register in Florida unless it is doing business in Florida. If all the corporation is doing is opening a bank account, that does not qualify as doing business in Florida.
See Chapter 2.1 and read the statutes it cites.
If they are doing business in Florida and you are requiring a copy of their authorization, all you are asking for is evidence that they have complied with state law.
Regardless, you may require evidence of a foreign corporation's Florida registration simply because you require it. By registering, the foreign corporation is submitting to the jurisdiction of the Florida court system. If they do not register in Florida and you want to sue them; e.g. major overdraft or other loan default, you need to go to the place (state or country) where they were established and sue them there.