Welcome to the site, Noland.
I'd say there is no reason not to use the RTS. Very good oil, and it's already paid for (sitting in the garage).
There is some theory that suggests that an overly thick oil might draw down the power slightly due to pumping loses, but that is just a VERY minimal issue here. I can assure you that absolutely no harm will come to your ZTR Kaw engine from using this grade oil. It's more of a topic for debate regarding fuel ecomony, which typically is not an issue when mowing grass, especially if you're only doing it as a homesteader and not commercially.
In the future, once the RTS is gone, you might consider a good HDEO 10w-30. I personally use Rotella in that grade (in my air cooled 19hp Kawasaki powered Scag ZTR), but certainly the Delvac and others would be appropriate too. My thought here is that (presuming again you're at home and not running commericially) you don't put enough hours on the unit per year to extend an OCI, and therefore the synthetic isn't needed.
But, since you already have the RTS, using it is a prudent and wise choice.
Conventionals vs. Synthetics isn't about which is "better"; it's about which lasts longer, while assuring safe operation, in relation to cost. Any product can be over or under utilized. The same applies to filters.
Make an informed decision; first consider your operating conditions, next determine your maintenance plan, and then pick your lube and filter. Don't do it the other way around ...