In theory ....
The Bank can only apply for a standard DBS check if the role is an FCA/PRA controlled function (see the link below for the list).
https://www.fca.org.uk/firms/approved-persons/controlled-functions

In general, these are going to be sales/client-facing roles, roles within the payments process and roles which exert a significant influence.  Even then, the bar is pretty high with regard seniority and influence of the role. For example, in one bank I know well anyone selling derivatives, investment or complex products will require a CF30 from the FCA but anyone selling vanilla products or a cashier in a branch require no FCA approval and therefore only a basic DBS.

One thing to be careful of are authorised payment institutions which are on the increase (remittances, FX payments, e-money, fintechs..) - technically, any managerial role in these firms can be legally subject to a standard DBS check.

Remember, the DBS certificate after the check is sent to the applicant and not the Bank - you will at least be able to see what has been disclosed before you pass it onto the Bank.  While the DBS are in the process of establishing an electronic process, this is still some time away (Unlock have a lot of information on their site about this).

Bottom line - I would apply for the role and channel your energy into ensuring you are offered the job.  You could always pull out down the line if anything adverse appears on your certificate (which the Bank won't of seen). Better to have a job offer and say no if necessary, than not get the offer and never know ....

Hello,
I am applying for a job within a financial institution and I am subject to their credit & criminal record checks. Although in a financial institution, it is not a FSA approved role. I believe that this means it will be a basic criminal record check (my convictions should be spent). Can anyone tell me if this correct?
 Thanks