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Just a (sort of) layman's comment: it seems to me that you need a characterization of surjective submersions that involves the implicit function theorem. If that's really the case, I'd guess that the result still holds true in the tame Fréchet category, thanks to the Nash-Moser implicit function theorem.
@Chris Gerig: I finally got time to think about how to transpose your argument to my context (Rob Kirby's book "The Topology of 4-Manifolds" was quite useful in this respect), and I believe it goes as follows: Since picking dual frames establishes a one-to-one correspondence between trivializations of $TM$ and those of $T^*M$, the same takes place for $V$ and $V'$ by means of the bundle projections $P$ and $\mathbb{1}−P$. This also yields a "doubling" in the trivialization change of $TM\oplus T^*M$ due to that of $V$ and $V'$, amounting once more to no change. Do you agree?
As mentioned by Igor, the terminology was introduced by Penrose in the particular case of Minkowski space-time ("Asymptotic Properties of Fields and Space-Times", Phys. Rev. Lett. 10 (1963) 66-68). When developing the procedure in detail ("Zero Rest-Mass Field Including Gravitation: Asymptotic Behavior", Proc. Roy. Soc. London A284 (1965) 159-203), he was already aware that compactness of the conformal completion was a rather special property of Minkowski space-time. However, the name stuck to many physicists because of the convenience of the conformal representation of Minkowski space-time.
Wordline's question concerns general linear connections, not just metric ones, I suppose. Proposition III.7.9 on page 146 (quoted in my comment above) does the job at the required level of generality, but the argument is essentially the same.
I believe that would be Proposition 7.9, page 146 of the book of S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry - Volume I" (Wiley, 1963). Moreover, the second connection has zero torsion and the same geodesics as the first. Notice, however, that such a question is not really research-level and hence more suitable for math.stackexchange.
@Chris Gerig: I've edited my question and its context to make them more precise. As such, I'm deleting some of my comments which are now properly incorporated into the question and addressed in your edited answer.
