Banks want to see that you are dependable and that your business is feasible. To confirm that, they want to see what ties you to Hungary. Having residency is a great way to prove that you are a serious client, but everything demonstrating that you have local partners, suppliers, clients, and establishments will improve your prospects at the bank of your choice.
The most convincing document you can present is a business plan, supported by contracts with Hungarian companies, and a rental agreement for an office, a shop, or a warehouse – whichever suits your business activity the most. If you cannot sign contracts, you might want to consider asking your potential partners for reference letters – since these are not binding documents, they are easier to acquire but also less effective when it comes to opening a bank account.