You're correct that biostatistics is likely the more lucrative public health degree. In answer to some of your questions (all based on my experience in grad school, ymmv):

I don't think it would be necessary for you to learn statistical software in order to get in - you will learn this as part of your program. In my program, anyway, the MPH in biostatistics was very rare, almost everyone got an MS.

I agree it's important to be clear that this field is the right one for you before committing to the degree. My suggestion for you would be to spend a year working in a research position (RA at a university doing some basic data analysis, perhaps) and try to get your feet wet with the type of work you would actually be doing.

I'd say take the time to get those prerequisites so you're both eligible for the MS and also to prepare you for the program. Biostat is not easy and if you don't have a strong background in statistics or mathematics it'll be a much more difficult program.

You should be able to get funding as an MS student so by taking the time to get a more competitive application together you should be able to go to a program that will fund your education 100%.