Experience needed to transition from undergrad to graduate school of two completely different majors
There is a big difference between a Masters program and a PhD program, and there is also a difference between a Masters program by coursework, and a Masters program by research.  Suitability of programs like these, and their entry requirements, generally depend on the specifics of the program and the entry standards of the university.  Having said this, I will try to give some general guidance on this matter.

Master of Statistics (coursework-based): For a student such as yourself, who has an undergraduate degree in a quantitative area outside of statistics (but with a small amount of statistics in that program), a Masters degree by coursework would be an obvious place to start.  A coursework-based Master of Statistics program is generally a good fit for someone coming over from another quantitative degree.  Indeed, these programs are usually designed for students who did not do an undergraduate degree in statistics, but have enough background in mathematics to be able to learn the subject more rapidly than would occur in an undergraduate degree.  (Student who already have an undergraduate degree in statistics may be able to jump this hurdle and go straight into a PhD program.)

Although it varies by university, entry into a coursework-based Master of Statistics program would not usually require an undergraduate degree in the same field, or any research experience.  It would usually be sufficient to have an undergraduate degree in a quantitative subject where you have done the core maths requirements (calculus, linear algebra, etc.) that would allow you to understand graduate-level statistics courses.  If this is a program of interest to you, my advice would be to make inquiries with the universities that are suitable to you, and find out the entry requirements for their programs.

Master of Statistics (research-based): Research-based programs generally have higher entry expectations than coursework-based programs, so for these programs it might be hard to get in (or to do the work) if you do not have a solid undergraduate education in statistics.  For a student like yourself, the more usual path would be to start in a coursework-based masters program first, and then progress to research once you have done enough courses to give you a good grounding in the subject.  In some masters programs there is a combination of coursework and then research.  The coursework comes first and then the research occurs when you have a solid grounding in the field.

Putting the cart before the horse: I am a bit surprised at your suggestion that you would pursue a research assistant position as a means of trying to get entry to a masters program.  You already have some experience as a research assistant, so what you are lacking is not this experience, but rather, coursework in statistics.  In your situation I would think that you could apply directly for course-work based programs in statistics, and this would then be a lead-in to later research programs.