Most schengen countries nowadays ask passport along with identity card plus photocopy so that they would be able to confirm applicant identity.
Also some embassies and consulates further requires family registration certificate registered with the relevant local authorities at your home country, to make sure application name and family name match accordingly.
Once your identity is confirmed, they would be matching your passport full name with your provided financial documents and bank statements. Moreover schengen embassy/consulate usually ask bank statement along with maintenance certificate confirming your full name signed by bank manager.
You have stated that you intended to apply to Italian embassy and they require all your documents should match your passport details are presumably correct.
If you have some extra time available, you could always amend your name with your local bank. Usually you will send a request for full name instead of first name and sometime they would update in their system. However some banks need you to open a separate account if you need a name change. The best possibility you could always get by visiting your branch immediately. Also you will have to show your full name in your covering letter as well as in occupation documents.
From my previous experience with different schengen countries, it would be highly unlikely they accept your application