I did it! With an even more unrelated background than you, I got into a PhD program in psychology this year (I had a BA in English and an MA in Spanish Lit). It took me a year and a half to prepare, I took the Psych GRE and 5 psych classes at a cheap local school, and I worked in 2 research groups as a volunteer (though one eventually started paying me).
You already have some research experience, which is great. The quantitative skills will also be a huge selling point for your application, because many applicants don't understand stats very well and it's integral to psych research. You should probably keep doing the research for as long as possible before applying. I think you honestly have a good chance at getting in.
ETA: I prepared for the Psych GRE by ordering 3 test prep books on Amazon and going through every single practice test. Quizlet is also great for flashcards organized by topic (social psych, neuro, history of psych). I also read a pop history of psychology to help root everything in my mind. I ended up getting a 770 (95th percentile) and had many of my interviewers commenting on my score, so it worked!
Also I should specify that I applied to clinical and counseling programs.