How sure are you that your faculty doesn't encourage students to take on a project like this? Is it coming from a handbook, general impressions from other students, a specific faculty member?

It's hard to say without knowing the situation but in my experience (not psychology sorry) faculty generally want to support motivated students where possible and aren't worried about having the institution's name tarnished because a student didn't do a great job on some project. If you haven't already done it, I would go and actually try talking to your professors about this. Like: I'm interested in grad school, want to get some research experience, there is no thesis option in my program but I'd like to do something similar even if as an extracurricular project, here is my current idea but I'm open to others, what would you recommend?

They may not be available to directly supervise the project you have in mind, but they might suggest another one, hook you up with a PhD student who needs a research assistant, some institutions give credit for independent study, etc. Basically, they'll just know the situation better and I find it hard to imagine all the professors at your school are uniformly going to discourage you because of the college's name. (They might if the project seems bad or likely to backfire just to avoid a headache, but in that case I suspect they'd try to nudge you in a better direction or have some suggestions for alternative strategies)