I'll try not to rant.
Necessity of a Transition Course?
The way I see it, you will need a transition course if the following applies:
- Your students start out with Calculus;
- Your school mixes math majors with others in Calculus (for size reasons or other).
For instance, at Vassar the typical major starts with linear algebra in the fall followed by multivariate in the Spring. That's already very "proofy" (certainly the way they seem to do it is!) and, as you can check, there is no proofs course on their catalog. I had a somewhat similar experience doing my UG in France.
You need #2, because if you can have dedicated Calculus sections for math majors, then as Pete pointed out, you can take them through a course that will teach them both Calc and proofs, but you wouldn't want to submit the general public to that course.
So to come back to the OP's question, I don't really think there is much of a choice between option 1 and option 2 because which one applies depends only on your public and material circumstances, not really on pedagogical choices.
What I'm really interested in
So the way I see it, if #1 & #2 hold, then you do need that transition, because the first romp through calculus has to be more computational, or you're really short-changing your students, both math-majors and not, at least in a typical course. So at some point, the students will need to transition, i.e. OP's model 1.
Now a related question is how do you help the students transition. And I have yet to teach my institution's proofs course, but I am very skeptical of these. At the very least, I can say that I've seen textbooks that were not promising at all: I like the logic and truth table bits, though where I am this would be covered in Discrete Math, i.e. before the proofs class. But some texts have: here is a chapter about how to do proofs in linear algebra, here is a chapter about how to do proofs in geometry, etc., somehow emphasizing the differences instead of the commonalities in proofs.
I should mention that not all textbooks are this bad; this semester, we're using the art of proof that seems a decent book. However, I think it's interesting that none of these books really seem to stand on their own: they are textbooks first and books second, when most of the books in my math library can be picked up and enjoyed whether you're taking a course from them or not.
To me, a proofs course remains a weird animal. I'd much rather ease the transition within a specific topic (e.g. Linear Algebra in my Vassar example). Also, learning how to write proofs is a long process, just like writing in general. The proofs class somehow sends students the message that there's an off-on switch: before this course you don't know how to write proofs, after this course you will; leaving a rather misleading impression.