"I propose that a person can be a Stoic Epicurean."
pshhh. I had that thought way before you.

8-)

But, honestly, they were opposing schools of thought, in that the Stoics believed that there is reason permeating the universe in the form of a benevolent Creator. He created the world such that, being virtuous is necessary and sufficient for Eudaimonia. Also, they believed that our ability to reason is special in that it is a fragment of the gods. So that, when we are rational and reasonable we are most like the gods.

The Epicureans believed that gods exist and that they created the universe, but that they don't give a whit about us. We are completely on our own. According to them, the best life (Eudaimonia) is one in which pleasures are at a maximum. So, not instant gratification, but using reason to help determine how to live a life of maximum pleasure, overall- spread out over the entire lifetime.

I think the 2 can be compatible, in that pleasure is important. And reason and virtue are important. Seneca actually praises some things that the Epicureans said (while trying to convince his friend that Stoicism is actually better/closer to the truth).

Perhaps they're compatible in that the followers of both were ultimately most concerned with wisdom/rationality/reason.

They're similar in some respects. Roman Stoicism, though, encouraged participation in public life while Epicureanism does not. And as has been noted, the Stoics believed in an immanent God necessarily involved in the universe, and that we partake in God.

I think the two have incompatible ends, with the most important difference being that Epicureans treat pleasure as intrinsically good and virtue as instrumental, while the Stoics hold precisely the reverse (with pleasure being 'choiceworthy' but not an end in itself).