Enkidu straight destroys him. The main reason Alcides was a challenge was due to the nemean lion pelt which worked against man made things Enkidu doesn’t have the same problem as Gilgamesh and skewers Alcides
The Nemean Lion's Pelt resisted Gate of Babylon because its a singularity which rejects human history. Jester who is also a singularity has stated he can be pierced by weapons made by a god for a god. They appeared in the same work with a direct comparison so I'm guessing Alcides is the same. Enkidu is a weapon made by the gods for the purpose of keeping Gilgamesh in line so he would get through the pelt with every attack.
Alcides hasn't shown anything to survive Enuma Elish although the fact that he was trying to bait Gil into using it suggests he does have some answer. As it stands he'll have to use his one time ressurection which I believe comes with unfortunate side effects. I'm not sure he has any way to put down Enkidu who can indefinitely self-repair from anything short of his core getting destroyed. Hydra poison doesn't seem like it would work on clay. Alcides could steal his Noble Phantasms but Enkidu is a divine construct and he rejects everything divine, so he would probably refuse to do that on principle.
Alcides was also most likely amped in his last fight with Gil since he absorbed Jack the Ripper's demon form although the outcome of the fight wasn't entirely certain before it was interrupted though it sounds like Gil had the advantage. I also need to see a full translation of it though.