Have you explored how carrying penalty affects other activities, particularly combat, beyond running, and have you noticed similar effects on fatigue?
They have similar effect. I haven't tested it systematically so not sure of the exact ratios (but likely they are exactly same as for running), but there are very notable differences in fatigue build-up from heavy duty work at 0% (tool on the ground) and say 2%.
More generally, it's worth noting that the fatigue build-up is NOT based flat on the percentage of encumbrance penalty, it's more fine grained than that. For instance, a character might have same rate of fatigue build-up at 0% and 1%, but if they pick some more stuff (while still remaining within 1%) they can start becoming tired more quickly.