Heres how I see it. Once you have a degree, you can apply for jobs with the GCSEs you have. I have straight Cs in my GCSEs. There were some grad roles that I couldn't apply for, but not too many. I don't think your current GCSEs would hold you back too much in determining what you could and couldn't apply for. However, HR see applications all day long. From their point of view, where they are looking to cut numbers, an application with a big hole where GCSEs should be is easier to cut.
As such, Id probably look to do some GCSEs where I was confident of getting A*-B. Personally Id go for about 5-6. I think on an application form this looks like a reasonable number rather than looking like you've done the bare minimum to get on to Access course or in to uni. Something like maths, english, double science, psychology, sociology.
However, you could look at it in a completely different way.
I don't know how long those GCSEs would take to study. You could potentially start an Access course in September if you weren't doing GCSEs. Studying GCSEs may put you another year behind. You could go straight to the Access course and then consider using that year further down the line for some other CV boosting work. Studying a course with a year in industry or volunteering or something. There are loads of possibilities.
I dont know what kind of roles are out there for psychology graduates related to psychology itself. I only have the example of one person I know. They didn't go in to a graduate scheme, it was just a direct job hire (which then gave them experience to go on to DClinPsy). However, my experience applying to finance based grad schemes is that almost all of them asked for GCSE results. As such, I think the people telling you that employers don't care about GCSEs are wrong. If they didn't care, they wouldn't ask. Perhaps these people have experience in psychology roles and therefore know better than I do, but they haven't stated that.
Whether you upgrade your GCSEs or not, you'll be able to get on to an Access course, and that should lead to uni. I guess you need to weigh up the time, cost, effort and how much extra impact you think GCSEs will have. Whether you think that time could be better spent elsewhere etc.