Sounds as if you have covered all the bases you can, and are well prepared to choose.
To your questions: yes that remark of Dieudonne' is exactly an explanation of why the chain rule is a completely natural result, properly formulated as a statement about maps. The Inverse function theorem also has a similar statement: If f is a smooth function from R^n to R^n, taking say 0 to 0, and if the derivative at 0, is invertible as a linear map, then f itself is invertible as a smooth map, on some neighborhood of 0. The implicit function theorem can also be stated: if the derivative f'(0) becomes a projection after a linear change of coordinates, then f also becomes a projection on some neighborhood of 0, after a smooth change of coordinates. (I strongly advise however also learning the usual old fashioned statement and applications of this theorem, for use in practice. In my case at least, after learning this abstract statement, I had no idea what Mumford meant when he said the implicit function theorem means, given an equation, you can use it to solve for some of the variables in terms of the other variables. )
In my day, there was no math 25. There was the 2 year 11/55 sequence, and the 3 year 1/20/105 sequence, for calculus, plus honors versions of 1/20/105. Math 11 "used" (i.e. recommended and totally ignored) Courant, later Spivak Calculus, Math 20 used a book like Angus Taylor, and 105 (or honors 105) used maybe Courant, or David Widder, while 55 was based on Dieudonne', or Apostol's Mathematical Analysis, later Loomis/Sternberg.
Apparently 25 was created as a middle ground between the two sequences, more sophisticated than 20/105, but less drastic than 55. There was also the changing situation in which virtually all strong high school students began to arrive with some calculus, whereas earlier very few had it available in high school.
AP calculus in high school caused a problem for teaching college math, since it creates a misconception in the students that they should be entitled to "advanced placement". But many high school "AP" students are not even prepared to skip one non honors calc course in college. If they try, one has these poorly prepared students in second semester calc, but not knowing the first semester well. This causes many good students to actually fail out of second semester or second year calculus. One way to adapt is by lowering the level of the college courses to match the weaker level of preparation.
"Advanced placement" also orients these students poorly, since they go from being honors high school students to taking advanced but non honors college classes. Thus their peer group is wrong.
At some schools, like UGa and Chicago, the old math 11 style "Spivak" course was kept for entering honors students, followed by an advanced version, based (at UGa) on Ted Shifrin's book:
https://www.amazon.com/Multivariabl...lus-Manifolds/dp/047152638X?tag=pfamazon01-20
Other schools like Stanford and Harvard, dropped the beginning Spivak course and plunged bright honors students right into a 55 style course, often with disastrous results. Rethinking these offerings perhaps resulted in the creation of math 25.
The physics major mentioned earlier, who started Harvard in math 25, had already prepared in high school with AP calculus followed by a proof based course from Marsden and Tromba's Vector Calculus book, augmented by an introduction to differential forms, (and taught by a college professor, i.e. me). He remarked that he "could not have survived" Bott's math 25 without the preparation in differential forms. (Check out the beautiful (advanced) book by Bott-Tu: Differential Forms in Algebraic Topology).
(If you choose or evaluate your courses based on the level of the professor, you begin to realize what it means to take beginning calculus as an AP course in high school, sometimes from someone who was not even a math major, but had only taken calculus in college.)
We have talked only about sources for advanced calculus, but I also recommend getting hold of a copy of Michael Spivak's Calculus, as a resource for one variable calculus, done right; or maybe better Courant, or Apostol's Calculus, since those books mention physics, unlike Spivak.
Here are used copies of both volumes of Courant, for under $20 total, a bargain.
https://www.abebooks.com/servlet/SearchResults?an=Courant&cm_sp=SearchF-_-home-_-Results&ref_=search_f_hp&sts=t&tn=differential and integral
Oh yes, and this discussion reminds of the third aspect of a college class: 1) professor, 2) syllabus, and 3) classmates! In terms of stimulating classmates, math 55b should probably be favored.
But looking again at the syllabi, that 25b syllabus is already quite extensive, i.e. rigorous one
and several variable calculus, and the profs are both stars. So, and obviously the choice is up to you not me, but 25b + 113 looks to me like a more reasonable learning experience, at least on paper. So I guess the only way to decide is to try them yourself and get a feel, as you said. Good luck!
By the way, the stuff on axioms for real numbers in the 25b syllabus is covered in Dieudonne' chapters 1, 2. (And it seems pretty ambitious to cover all of the foundations of real numbers in just one 25b lecture, unless it is review.) If you can read that and maybe also chapter 3, you can likely handle anything theoretical you will see in these courses. I.e. Dieudonne' is the hardest source to read of all I mentioned. Here is a used copy for around $32. This is an amazing book for content, but very demanding reading (no pictures!).
https://www.abebooks.com/servlet/Se...Results&ref_=search_f_hp&sts=t&tn=foundations