I switched gears in a radical way after several years working in my Bachelor's field, by taking the additional classes I needed, primarily at a community college. Then I started graduate studies in a strong department in my new field. The professors didn't distinguish between me and fellow students who had gone to bigger name schools for undergrad. Chances are they didn't even know. To them I was just another student, interested in learning, asking good questions, working hard.
The admissions committee members would have known, but I never knew which professors had sat on the admissions committee. But even the ones who had looked at my transcripts (whoever they were) might have forgotten by the time I took a class from them.
I only had a problem with one professor. He assumed I didn't know my stuff when he was grading a qualifying exam. I think the root problem was that he was the one professor in my sub-field that I had not taken a class from. So to him I was a nobody. I found this out when I asked to see my exam and how it was graded. I discovered that he had graded it rather subjectively.
Thus, the one thing I would do differently if I were to go through it all again would be to take a class from that one guy. He only taught one class. I sat in on a couple sessions at some point, and found his teaching deadly boring, with very little homework. So I would have had to go to his office hours and really work to find something to talk about.
But that's what I would do differently if I could. Note, that had nothing to do with the schools I had been to prior to grad school.
Note that as a former intern, you will be a known quantity to the stellar company. That can be a real foot in the door.