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This is a rewording of Matthew's answer, but one can use nonstandard analysis to obtain a category that achieves what Mark wants. After taking ultrapowers, one obtains non-standard integers (ultralimits of standard integers), nonstandard finite dimensional Banach spaces (ultralimits of standard finite dimensional Banach spaces), nonstandard linear transformations (ultralimits of standard linear transformations), etc. One can then separate these non-standard objects into various classes, e.g. bounded non-standard reals (ultralimits of uniformly bounded standard reals, or equivalently non-standard reals that are bounded in magnitude by a standard real), bounded nonstandard linear transformations, poly(dimension)-bounded poly(n)-bounded nonstandard linear transformations, polylog(dimension)-bounded polylog(n)-bounded nonstandard linear transformations, etcetc., where n is an unbounded nonstandard natural number (which, in practice, would be used to bound dimensions of things). Each of these form a category.

Thus, for instance Kashin's theorem becomes the statement that every nonstandard finite dimensional Banach space of some nonstandard finite dimension N has a half-dimensional an M-dimensional subspace which is isomorphic (in the category of bounded nonstandard linear transformations) to $\ell^2$.\ell^2(M)$ whenever $M \leq N/2$ (or more generally when $M \leq (1-\epsilon)N$ for some standard $\epsilon > 0$.

Dvoretsky's theorem is trickier. Here, I guess one needs to work with the category of almost isometriescontractions: operators whose operator norm, and norm of inverse, is at most $1+o(1)$ (i.e. between $1-\epsilon$ and bounded by $1+\epsilon$ for every standard $\epsilon > 0$. Then I think the theorem says that any nonstandard finite dimensional Banach space with some nonstandard dimension N has a M-dimensional subspace that is almost isometric to $\ell^2$ with dimension at least M, \ell^2(M)$, whenever $M = o(\log N)$. (I may have messed up the quantifiers slightly, but this is pretty close to what the nonstandard translation of things.things should be.)
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This is a rewording of Matthew's answer, but one can use nonstandard analysis to obtain a category that achieves what Mark wants. After taking ultrapowers, one obtains non-standard integers (ultralimits of standard integers), nonstandard finite dimensional Banach spaces (ultralimits of standard finite dimensional Banach spaces), nonstandard linear transformations (ultralimits of standard linear transformations), etc. One can then separate these non-standard objects into various classes, e.g. bounded non-standard reals (ultralimits of uniformly bounded standard reals, or equivalently non-standard reals that are bounded in magnitude by a standard real), bounded nonstandard linear transformations, poly(dimension)-bounded nonstandard linear transformations, polylog(dimension)-bounded nonstandard linear transformations, etc. Each of these form a category.

Thus, for instance Kashin's theorem becomes the statement that every nonstandard finite dimensional Banach space has a half-dimensional subspace which is isomorphic (in the category of bounded nonstandard linear transformations) to $\ell^2$.

Dvoretsky's theorem is trickier. Here, I guess one needs to work with the category of almost isometries: operators whose operator norm, and norm of inverse, is $1+o(1)$ (i.e. between $1-\epsilon$ and $1+\epsilon$ for every standard $\epsilon > 0$. Then I think the theorem says that any nonstandard finite dimensional Banach space with some nonstandard dimension N has a subspace that is almost isometric to $\ell^2$ with dimension at least M, whenever $M = o(\log N)$. (I may have messed up the quantifiers slightly, but this is pretty close to the nonstandard translation of things.)