A bit of history question - slogan "Bumper to the Flag"?

"Bumper" is a word with multiple meanings. One meaning, now out of fashion, was a glass or mug filled to the brim with alcoholic beverage, to be use in a toast with companions holding similarly replete glasses.

So, "Bumper to the Flag" (or "Bumper to" ... just about anything one chose) was a way of saying: "Here's a toast to the Flag!"

I do not know when, or quite how, the word "bumper" came to be used in this fashion, but I speculate that it has to do with the happy habit (retained to this day) of clinking (or "bumping") glasses with one's companions immediately prior to toasting the health of some symbol or person.

I am reasonably certain that the word "Bumpers!" was used to invite such glass-clinking, in the same way that nowadays we might say "Cheers!" or "Bottoms Up!". Why do I believe this? Well, Patrick O'Brian, in his wonderfully well researched and beautifully written Aubrey-Maturin novels about life at sea around 1800 AD, uses the expression "Here's Bumpers to ____" frequently in his description of drinking bouts at dinners.

This explanation seems to fit with your finding the inscription on drinking vessels commemorating the Civil War.