Should I push for my paper to be published as an undergraduate?
Perhaps your advisor is worried about the depth or novelty of your work, suggesting that it might not be worthy of publication. s/he might be correct or not. But there is no reason to restrict yourself to an undergraduate journal. If the work has the right quality it can be published anywhere that the referees and editor will accept it. The fact that you are an undergraduate has no bearing on the issue of quality. 

You can submit a paper, certainly, though whether as a sole or joint-author depends a bit on your field and location. It might be worth it just for the experience. 

But speak again with your advisor/mentor to find out why they were originally a bit dismissive. And look at what appears in any journal you might submit to in order to judge your paper against what they normally accept.