I agree that both have the same effect in the end on would-be uncontrollable passion, but when two things act in different ways toward the same end, is it not good to employ both to make doubly sure that end is reached? For example, if you really want to make sure your counter top was germ free, you could use both an antibacterial cleaning agent, and a sterilizing lamp. As long as the lamp did not cancel out the effect of the agent or vice versa, this would seem to be practical. Perhaps this is what you mean below…
But aren’t you using the word “emotion” in two different senses in the above? It seems to me, we have a few concepts here that need clear delineation…
Emotion:
I don’t think Stoicism would reject all of the things we commonly think of as “emotion”. When Stoicism deals with emotion, it is referring to great emotional stirrings that would make us suffer or act out of control. So, we must be clear which version of this word we mean.
Compassion:
Some would describe this more as a set of priorities and values on which we act, rather than an emotion (or at least, rather than MERELY an emotion). In either sense, I’m sure that a sensible version of compassion is not what is meant to be excluded by Stoic thought. Therefore, the expansion of it through Buddhist methods should be compatible, as long as we’re not talking about a compassionate emotion so out of control that it leads us to do unwise things. I don’t think Buddhism would advocate that either.
Passion:
In one sense, perhaps the more common, this means doing something with conviction which includes emotion. Stoicism would then not exclude every version of this. However, in the sense that Stoicism uses the word, it means (as with “emotion”) an uncontrollable stirring of the soul which leads us into folly and even evil. So, by the common conception of the word, the Stoic use would be referring to a specific subset only. We must be clear which of these two versions we are referring to.
One thing I think would be a mistake, would be to find a word (for example, “emotion”) in a paragraph from Stoic literature and Buddhist literature, and then think that they are talking about the same thing and must therefore be either compatible or incompatible based on that sort of thing alone, rather than looking at the concepts behind the words.
In short, if we are careful about what all of the definitions mean precisely in (1) Buddhism, (2) Stoicism, and (3) common usage, then it seems to me these two philosophies might be simultaneously practiced with consistency and maintaining their integrity - although I admit I still have much to learn about each. 
I think what might convince me otherwise are examples where the substance of what is being said in one school is directly contradicted by the substance of what is being said in the other. This would take the form of either an irresolvable and important difference in conception of reality, or an irresolvable and important difference in how we should think or behave. By this, I don’t simply mean differences, but logically exclusive differences.
If I say, “Fred’s coat is red” and you say “Bill’s pants are green” these are different, but not irreconcilably so. I haven’t seen Buddhism say “Fred’s coat is red” and Stoicism say “Fred’s coat is blue” yet.