New idea:
The selection of a catalyst not only determines what Servant is summoned, it also determines their abilities, based on how the catalyst is connected to their legend.
The ancient snakeskin is connected with the legend of Gilgamesh, unfortunately, it is connected as a part of the Golden King's greatest failure. Snakes gained the ability to shed their skin when a snake got into his pack and consumed the herb he needed for his quest for immortality. Thus, it is a symbol of the King of Heroes' inability to securely pack stuff.
As such, Gilgamesh has no control over the Gate of Babylon. Yes, he has just about everything in there, but he can't reliably find anything. If he tries to remove a single object, it will almost certainly not be what he's looking for. The only surefire way to get what he wants out is through the blade-spam method, purely because if he dumps everything out, the item he wants will show up sooner or later, assuming he sees it in the rain of weapons, baubles, ancient Sumerian cookware and other miscellaneous junk.
I'm sure other people can think of artifacts that reflect negatively on the hero being summoned, and how it might amusingly impact the summoning.