Well, I used some bribery to disrupt trade relations and start wars.
I also used to spies to perform civics changes to eliminate those 'we like your choice of civics' bonuses.
If anyone of my allies ever joined gilgamesh in a war against another civ, I quickly got them out so the mutual military struggle didn't build up.
I did everything I could to make sure Gilgamesh got to be second in population (which I unknowingly failed at) as well as increasing my positive relations with allies and doing everything possible to reduce positive relations between my allies and gilgamesh.
So, in all ironies, things couldn't have worked out better at the end of the game. Hannibal had vassalized Pacal while Hatty and Monty were Gilgamesh's vassals. The turn I built the UN, Gilgamesh declares on Pacal, which means he also declares on Hannibal (master). Gilgamesh's vassals (Hatty, Monty) also join the fray. I brbe Bismarck and Toku to go to war with Gilgamesh (and thus Hatty and Monty too). I thought it had sealed the deal until I saw my secretary general vote come up against Toku.
Oh well, global armageddon is just as fun. I just completed the Manhattan Project and will be gifting Uranium and fission to all my allies. If you can't save the world with a diplo victory, might as well watch it burn.