I’m not clear from your question whether your difficulty relates to opening a bank account or obtaining merchant services. Banks are required to follow the guidance contained in the Banking Code.
Applying to a bank for a current account for a business need not require the bank to extend overdraft facilities or to offer a loan.
However, what is appropriate will vary from case to case.
Your credit history might well be a source of information the bank will access to make its decision.
The bank would need to be clear of your partner’s involvement in the business.
Dependent on the business you have started and the trading situation to date they may ask to see a business plan to assess the likely conduct of the account in the light of your existing commitments.
Forming a limited company is unlikely to assist in this situation as the business is unlikely to have traded for sufficient time to have built up any assets. The bank will look at the track record of those involved in the business.
If you are applying for merchant services the bank will asses whether they are able to meet the business card processing needs.
This will involve looking at business plans and accounts, the businesses cashflow, profitability and other commitments, other personal commitments and information from credit reference agencies.