Way to get the Reeking Cave Unusual Gem before the associated main quest

I want to get Prowler's Profit early in the game and I've been ignoring the main quest-line. In fact, I'm still on the quest after Unbound. I'm running the latest patches (and unofficial ones), so the Unusual Gem must be in the Reeking Cave. It seems the Reeking Cave can only be accessed (without Whirlwind Sprint) through the Thalmor Embassy, but the compound is walled off and the front gate is locked and can't be picked. There doesn't appear to be anyone outside of the walls, and I've read that nobody has the key anyway. I've tried coaxing some guards to come at me, but they won't open the gate.

Is it possible to get this Unusual Gem without Diplomatic Immunity? If so, how?

Update: I did find a way in, but I warn that it may be futile. I acquired the quest by reading the book in the chest, which is the only way to spawn the guards that are supposed to hold keys to the Reeking Cave door (because the other trigger doesn't spawn), but neither of the guards were holding keys to it. What a wild goose chase.