I believe you are right on your diagnosis. Good job!
Running can cause stress on the gluteus maximus and piriformis musculature.
With each heel strike, there is a normal internal rotation of the femur, whose motion is controlled by the piriformis musculature; overtraining may cause injury to the piriformis.
Running uphill will cause forward flexion of the pelvis, which in turn may cause a strain of the gluteus maximus musculature (which normally posteriorly rotates the pelvis).
Running injuries leading to sciatica like symptoms can be avoided by changing sneakers every 250-400 miles, at which point the sneaker loses 40% of its shock absorption capabilities.
Initiating a good flexibility and strength program for the lower extremities would help in the prevention of repetitive strain injuries such as sciatica.
Biking with the pelvis flexed too far forward while reaching for the handlebars can also strain the gluteus maximus musculature.
If the handlebars are set too far below the seat level, the rider may be forced to lean too far forward.
A proper bike fitting may help prevent this injury. Be sure to consider bike size, seat height, and handle-bar height/angle.
You might benefit from putting ice on the area 20mins on 20mins off for 1 hour 1x a day.
I know this sounds weird but a deep tissue massage ( better have your wife ok that one) could help.
Tylenol might help since the nsaids are rough on your stomach.
If this persists you might want to see a Sports Medicine Physician.
You are welcome and thanks for the bonus!
Hope you feel better soon!