I would not ask about the chances of being published. Given that you have no experience, it's almost insulting to ask about what you can do to become published. If you really want to get published, show your PI and research mentor your motivation and work ethic. If you put in the time to understand what you're contributing to, the time to read the literature, the time to actually run all the experiments, the time to take care of your cell lines (or whatever other model you're using), you have done everything you can to succeed to lab. Publishing is a slow and painful process.
The manuscript I am authored on has been through two submissions with revisions in between, with the PI currently rewriting for submission to another journal. Not to mention other publishing drama from competing researchers. It has been at least 6 months since the paper has been "finished" and "ready for submission." If you come in thinking you'll get published, you'll likely be disappointed and frustrated.
Not sure OP has the luxury of choosing from multiple opportunities. I am certainly not going to dig into my PI's personality to decide if I want to LEARN from him/her and GROW in the environment. Let his credentials and publication record back that up. Yes, a good mentor is a huge boost to your first experience, but if you are truly determined to become published, it should take more than a bad mentor to ruin that ambition. Most PIs have had their fair share of nonexistent, mean mentors. They are faculty members now because of their talent AND their work ethic.
And just from personal experience, I hate seeing other undergrads stroll into a lab with the intent to get their own project/publication. By the third month, most of them haven't even read 5 papers and are stuck doing ONE thing over and over.