"I notice that bands on Bing pipes are not always the same size, and sometimes, his pipes have no bands. Maybe he liked having variations on the same style of pipe?"
No, I do not believe so. Rather, my best guess is that given his fondness for the Merchant Service pipe, and due to the small diameter of their shanks, many of the bands were applied (possibly at different times by different craftsmen) as repairs to breakage.
As an interesting sidelight, notice that the pipe pictured in the image above features a fishtail stem. According to Richard Carleton Hacker in his book titled "Rare Smoke - The Ultimate Guide to Pipe Smoking" this was a rare option. He also stated that the pipe could feature a factory-applied silver band, and shows an image of one. That band is much shorter than the one shown in the image above.