All I know is apparently the "parts guy" don't know **** and I would stay away from going to him for advice.
Also his comment that you don't "need" synthetic unless your towing heavy sounds like old school foolish logic as well. I get that everyone has different opinions and high end synthetics aren't always a "need." But fluids aren't where I look to save myself money, especially when considering some of the high end synthetics last so long that they actually come out to be the same price or cheaper then cheap stuff all while giving you better protection the entire time.
But like posted, you will be fine whether you have an open or limited slip since the additive is in it, I didn't see you post what you had. I wouldn't kill myself to go change the fluid super fast; id atleast get some use out of it before dumping for synthetic and that is coming from a huge synthetic guy. Just my 2 cents. If you are open diff you don't need additive but factory limited slip you do. But keep in mind that many of them have it already included with the gear oil, it just depends on what you buy.
If you go aftermarket set ups, usually it is best to check with the manufacturer of it because they occasionally don't recommend certain additives and such for their products.