Issues with BPSS check. How do I proceed?

I’ve recently been offered a job that requires a BPSS check, however I’m not sure how to proceed due to two potential issues.

• It asks for employment history over the last 3 years, there were two jobs I left within 1 week due to not being a good fit and decided to leave those off my CV as I didn’t think they were relevant to my experience, should I include those jobs in the employment history section and will including them throw up any issues due to leaving them off my CV?

• The job asks for proof of identity, however I have recently changed my first name via statutory declaration and am still waiting for my new passport and driving license to arrive in the post, I’m also still waiting for HMRC to fully process my name change. How do I go about confirming my identity?