I'm not Halcyon, however I have some experience with the "standard" and smaller sized backplates, because I am 5'5" (but very short-waisted).
To me, it's all about your torso length. If you have a long torso/shorter legs, then maybe a standard plate would fit you; if you have an "average" torso length for your height, or a shorter torso and longer legs, then I would imagine the Small backplate would fit you better.
My first exposure to a backplate was when I rented a Halcyon Eclipse for two weeks. They only had the "Standard" sized backplate, and it was definitely too long on me (short torso). When I had the waist band around my waist, the top of the plate was too high. That caused the top of the plate to be able to shift around because of how the plate came up too high and so the shoulder straps didn't lead very well. I did dive it for two weeks, but I would not have bought it.
I was able to try on the "Small" plate and it did fit me much better (although they didn't have a Small for rent or purchase at that time). As a male you may have a bit more flexibility vis-a-vis dropping the waistband to get the top of the plate down.
I actually ended up getting a 13" plate (Halcyon's small is 14"), and it fits me perfectly even though I'm 2" taller than you, simply due to my torso length. I could see other people my height potentially fitting into the "Standard" (15.5") plate, if they had proportionally very long torsos.
All that to say that if I had to guess, I would say the Small Halcyon plate would fit you better, but that it's torso-length dependent. (Also the small plate is slightly narrower, but that may not be as important.)
I have to wonder though: Is the LDS convincing you the "Standard" backplate fits you because they don't have a Small in stock? Or have you tried both on and you are still not sure? If it's the former then I, personally, would want to try on the Small before investing in the BC. If it's the latter then have a look at this page for a visual sizing guideline:
BAUE How To Size a Backplate
Blue Sparkle
PS: One note that may affect you as well: The Halcyon Small plates (at least the ones I have seen) do not have cam-band slots, so they necessitate the use of an STA with single tanks. (Not that there is anything wrong with that; it's just a difference from their "Standard" plate.)