I can only help a bit. First, there were a number of 11mm cartridges, from 1870 into the 20th c. Most of the North Sea neighboring nations would be using them in their armies: 11mm Gras, for France, 11mm Mauser, for Germany, 11mm Danish, for Denmark, 11mm Beaumont, for the Netherlands...and there would be many similar 11mm ones, in other countries ( in part because many of them bought the same Remington rifle). They were black powder, low-pressure cartridges, and after around 1890 they were all replaced with smokeless-powder smaller-caliber ones- like 8mm Mauser. 11mm cartridges continued to be used for hunting, however, and with smokeless powder were very powerful, used for hunting big game like elephant. These also saw a little use in WW1, and there was a special one made from the old 11mm Gras that was used in a Hotchkiss machine gun for shooting at balloons.

You have a cast lead bullet , with lubricant grooves- something typical for black powder. But it has the remains of a copper gas check on the back- that's something that would be used for soft lead bullets with smokeless powder. It also has a spitzer point, not the rounded nose you would expect of a normal soft lead bullet, certainly the normal bullet for hunting. And, no, it does not look like it was used in the Hotchkiss Balloon Gun. As a cast lead bullet, it could have been made by anyone with a mold.

So, unless a collector comes forward who has seen these before, there's only speculation. Did the Germans arm some of their Volksturm with old 11mm Mauser rifles and hand-cast bullets, in 1945? Did somebody shoot at seals with his elephant gun in the North Sea, and make his own bullets? You have a good mystery, there.