I've done the Katy end-to-end 9 times and one of those in 2020 was on a tandem! We have a newer Schwinn Twinn MTB tandem which is very heavy plus the stoker and I aren't lightweights. We used 26x2.125" tires with some knobby tread and it worked well on the Katy. We also rode the GAP and C&O trail last year and we were very thankful to have the knobby tires on the C&O as it was muddy in places.
Last year I did the Katy solo on a gravel bike with 650B x 47mm tires which I thought were overkill. However, they were smooth with just a tiny bit of tread. One day out near Clinton it had rained hard the night before and I did have trouble slipping and sliding. I'll be putting on some mild knobby tires the next time.
I've only had three flats on the Katy Train in thousands of miles of riding on it. Still puncture resistance is a good thing. Also the Katy is hard packed around Columbia and St. Charles where it's ridden a lot. However, if you get out into the boonies, it can be mushy at times.
The Gravel Kings seem to have good enough tread. I'd recommend the largest size that will fit your tandem.