And finally: I assume the general (non-Brownian) prescription for the affine subspace is $f(t)\psi^{(t)}/ \psi^{(t)}(t) + \mathcal{B}_0$? And might you have some reference for this construction?

Great, thanks—so, in particular: 1. The CM space is the same for any conditioned or unconditioned process of the same "type". 2. Conditioned processes in general no longer live in a Banach space, but in an affine subspace of a Banach space. 3. $\mathcal{B}_0$ is independent of the pinned value and depends only on the "time" of the "pinning". 4. In constructing the affine subspace, the particular sample path $f$ is irrelevant, only its value at the "pinning time" matters, since it is "replaced" with the scaled Riesz representative of the evaluation functional. Is all of that correct?

Given my later additional realization, maybe I should rephrase the whole question, but: I want to know how processes whose final (or initial, similarly) value is pinned to a non-zero value are treated. If the pinned value is not zero, then the sample paths no longer form a vector space, so it must differ from the procedure for, say, the Wiener process. If I'm understanding you correctly, then the answer is that one says that there is some Banach space E and the sample paths are given by E + (b-a)τ/T? Is this the general construction?