I am trying to prove the following statement of real Bott periodicity, on the level of actual spaces of Clifford module extensions (i.e., not equivalence classes of modules).

Let $W = \mathbb{R}^{\infty}$ (in the direct sum sense), equipped with the standard inner product. Fix in advance a set of skew-symmetric operators $e_1, e_2, \ldots $ on $W$ for which $e_i^2 = -I$ and $e_ie_j = -e_je_i$ for $i \neq j$. Let $X(n,W)$ be the space of operators $f_n$ on $W$ which anticommute with $e_1, \ldots, e_{n-1}$ have $f_n^2=-I$, and for which $\ker(f_n-e_n)^{\perp}$ is finite dimensional, and equip this space with the topology induced by the operator norm. I wish to show that $$ X(n,W) \cong X(n+8,W) $$ where $\cong$ denotes homeomorphism, or barring that, homotopy equivalence. Note that $X(n,W)$ is the space of orthogonal $Cl_n$-module structures on $W$ which restrict to the "standard" $Cl_{n-1}$-module determined by $e_1, \ldots, e_{n-1}$. Here $Cl_n$ denotes the real Clifford algebra on $n$ generators with negative definite quadratic form.

I hope to make use only of the well-known isomorphism of real Clifford algebras $Cl_{n+8} \cong Cl_{n}\otimes_{\mathbb{R}} Cl_{8} \cong Cl_{n}\otimes_{\mathbb{R}}\mathbb{R}(16)$. I'm aware that additional technical assumptions may be necessary, for instance regarding a "complete universe" of representations, but my hope is just to get the basic idea.

I would be content to show that the space of irreducible $Cl_n$-module extensions on $V$ of appropriate dimension is homeomorphic to the space of $Cl_{n+8}$-module extensions on $V \otimes \mathbb{R}^{16}$, with the map induced by tensoring with the canonical representation of the $16 \times 16$ real matrix algebra $\mathbb{R}(16)$.

The main paper I have been following is Behrens - Addendum to "A New Proof of Bott Periodicity".