I don't know how accurate this is but

If this is a LEED building the shower will likely use 2.5 gallons per minute, but more likely 1.8 gallons per minute. The default setting in the WEc3 template is a 5 minute shower duration. Most showers are a mixture of hot and cold water. I don't know what percentage. I can guess 75% hot and 25% cold.

1.8 gpm * 5 minutes * 440 BTU * 75% = 2,970 Btu.
Compared to your 16,667 Btu value for the 4 mile car ride.

There is embodied energy in the shower construction, plumbing, pumping, and also bike construction. And in the car's construction. The best way I know of to estimate this total energy cost is the total dollar cost to construct and maintain.

I just wrote out a detailed response with my own calculations but then lost them with an errant keystroke! Drat.

My quick calculations, based on kwh, showed about 2.4 kwh for heating the water -- and just a wild guess of that same about for potable water provision and subsequent treatment -- so 5 kwh. And, 24 kwh for a 10-mile commute. I think the shower comes out of top.

Question -- most people would take a shower at home anyway, so showering at work replaces that. Many people would also shower when they get home anyway. Probably a lot of people will shower after returning home on the bike. So, it's not definitive whether bicycling adds overall showering. Probably it does on average, but not by as much as you imply in your question.

A few other factors:

-the Btu (and water and possibly chemicals, if not organic) involved with growing the additional food one must consume to commute by bike
-the enormous difference in paved area (heat island) required for a car than for a bike, and the saved infrastructure as a result of widespread bike commuting
-the lower air pollution from fewer cars on the road and reduced obesity problems resulting from biking or walking to work, so Btus saved from less hospital infrastructure

I know this is all theoretical, but it's not exactly apples to apples.