I'm in psych. I considered going for school psychology programs, but ultimately decided to do health psychology instead. In any event, this is what I gleaned from looking into school psychology programs at the time.
I don't think school psychology programs would wonder why you want another master's when you already have one. School psychology is a practice degree; you can't practice with a writing MA, and it's understandable that you'd want an MS in school psychology so you could actually practice school psychology.
As a side note, Ph.Ds in school psychology often work within schools and do school practice. It's considered a scientist-practitioner degree and many of those applicants are gearing up for working within schools or as administrators of psychological programs in schools. I don't think you have to explain why you don't want a Ph.D, just talk about why you want an MS (because you want to work with kids and practice in a school-based setting). Lots of people get more than one master's degree.
You actually may look to apply to Ed.S programs in school psychology (specialist's degree) -- the pay is better, and in some states you may need an Ed.S instead of an MS to practice. Visit
www.nasponline.org for more information (the website to the National Association of School Psychologists); their page is very informative. Only specialist level and doctoral level programs in school psychology are approved by the NASP, although I know in many states you can use just an MS to practice.
Also, don't say "I'm just not a good standardized test taker." It's defeatist. You CAN study for the GRE and raise your score -- you just need to practice and learn the strategies. I used to tutor the SAT and the first thing I always did was forbid my students from saying "I'm just not good at taking tests!" It's not a natural talent; it's something that has to be learned and sharpened. People with high scores on the GRE didn't come out of the womb knowing how to do it. (Sorry, I put on my tutor hat for a second there :D)
I think your total GRE score (1140, if I added correctly) is fine for master's programs, but you might want to try to get up that quant. Psychology programs are one of those fields that value scores in both areas, because so much of psychology is quantitative.