I have read Spivak's Calculus and it has went well. I didn't have any problem with the rigorosity of the book at all.
Now, I have never had any experience in multivariate. I only have experience with basic high school calculus, linear algebra and the calculus from Spivak. Can I learn multivariate calculus directly from the Baby Rudin analysis textbook (does it cover that)?
On
Having gone from Spivak's Calculus directly to Rudin's Principles myself, I would say the transition is fairly smooth. It would be ideal to have seen multivariable from somewhere by chapter 9, but up through chapter 8 there is no real need for that sort of material.
On another note, I recommend finding another reference for multivariable than Rudin; starting from chapter 9, my opinion is that his exposition gets considerably and irreversibly messier. Try to find something that has a lot of more computational exercises (the kind of book that a non-honors multivariable calculus course might use at your university, perhaps); multivariable calculus is valuable, and deserves a lot of practice.
This is less of an answer, and more a recounting of my personal experience.
I went through baby Rudin without previously having taken Calc III, and the result is that for me, whenever I want to understand something in 3-d calculus, I think in terms of $n$ dimensions, and then specialize to $n = 3$. For instance, I couldn't tell you the low-dimensional Stokes's theorem or divergence theorem in the formulations involving normal vectors and such off the top of my head.
I don't know anyone else who has done this, but I suspect that anyone going through the multivariate part of Rudin without outside reading would develop similarly idiosyncratically.