The natural generalization of your "straightest-possible" constraint is a curvature-constrained path.
I know this is not what you asked, but I wanted to mention that even finding a path with maximum
curvature 1 inside a convex polygon is already difficult. This was studied by
Agarwal et al. in a paper,
"Curvature-Constrained Shortest Paths in a Convex Polygon,"
(SIAM Journal on Computing, Volume 31, Issue 6, 2002)
Here is one of their figures, which hints at why it is difficult to find a path from between initial $I$ to and final $T$
Returning to the question (or questions) you did ask, I think looking at the paper "Polygonal path simplification with angle constraints," by Chen et al. (Computational Geometry, Volume 32, Issue 3, November 2005, Pages 173-187), might help. They explicitly consider the "tunnel" version of your revised question, phrased in terms of an error tolerance $\epsilon$. Perhaps most usefully for your purposes, they describe all the related work in this domain, which has considered many variations. Usually those variations start with a path $P$, and then try to find another path (a "simplification") with certain properties: (1) The min-# problem: Find a path that remains within $\epsilon$ of $P$ but has the fewest segments; (2) The min-$\epsilon$ problem: Given a fixed number $m$ of segments, find a path using no more than $m$ segments that minimizes $\epsilon$. This specific paper solves the min-# problem with the addition of angular constraints, which seems close to your initial formulation (as clarified in the comments).
Edit. Here is a preliminary version of the "Simplification" paper: simplification.pdf.