About 15 years ago I used Karoubi's description of $K$-theory to solve prove some cutting and pasting formula for the index families of elliptic problems. To do so I needed rephrase Karoubi's theory into something more flexible and more computable. Some of the interpretations I found might be relevant to your question. I will briefly describe one such interpretation referring for proofs and many more details to the original source, my old paper Generalized symplectic geometries and the index of families of elliptic problems, Mem. A.M.S., vol. 128, no.609, 1997. I will refer to this as the old paper. $\newcommand{\bsH}{\mathscr{H}}$
To simplify the presentation let me define a $C^{p,q}$-module to be a Hilbert space $H$ equipped with a $8$_morphism of $C^*$_algebras $\phi: C^{P,q}\to B(H)$. A graded $C^{p,q}$-module can then be identified witha $C^{p,q+1}$-module. $\newcommand{\eF}{\mathscr{F}}$
Suppose that $H$ is a $C^{p,q+1}$-module. Define $\eF^{p,q}$ (or $\eF^{p,q}(H)$ to be the space of closed, densely defined, Fredholm, selfadjoint operators $T: H\to H$ that super-commute with the $C^{p,q}$-structure, i.e.,
$$ Te_k+ e_k T=0,\;\;\forall k=1,\dotsc, p+q. $$
The space $\eF^{p,q}$ carries a natural topology defined by the metric
$$ d(T_1,T_2)= \Vert \Psi(T_1-\Psi(T_2)\Vert,\;\;\Psi(\lambda=\lambda(1+\lambda^2)^{\frac{1}{2}}. $$
Denote by $\newcommand{\eBF}{\mathscr{BF}}$ $\eBF^{p,q}$ the subspace of $\eF^{p,q}$ consisting of bounded operators. Then one can show (see here) that the inclusion $\eBF^{p,q}\hookrightarrow \eF^{p,q}$ is a homotopy equivalence and that $\eBF^{p,q}$ is a classifying space for Karoubi's $KO^{p,q}$, which for simplicity I will denote by $K^{p,q}$.
Thus, to a compact $CW$-complex and a continuous map $T: X\to\eF^{p,q}$ one can associate an element $(E,\eta_0,\eta_1)\in K^{p,q}(X)$. To effectively describe this correspondence
$$ (X\stackrel{T}{\to}\eF^{p,q}) \to (E,\eta_0,\eta_1), $$
one needs a new, symplectic description of Karoubi's $K$-theory, and it is through the symplectic prism that I got to see Kasparov's KK lurking in the background.
Let $T\in \eF^{p,q}(H)$. Recall that, by construction $H$ is a $C^{p,q+1}$-mpodule. The direct sum $\hat{H}=H\oplus H $ has a richer structure of $C^{p+1,q+1}$-module (see section 5.2 in the old paper)
The graph $\Gamma_T$ of $T$ is a closed subspace of $\hat{H}$. Denote by $R_T:\hat{H}\to\hat{H}$ the orthogonal reflection in $\Gamma_T$. Observe that the subspace $H\oplus 0\in \hat{H}$ can be identified with $\Gamma_0$, the graph of the trivial linear map. We set
$$\Gamma_\infty=0\oplus H\subset \hat{H}. $$
Then $R_T^2=1$, and the condition $T\in\eF^{p,q}$ is equivalent with the following requirements.
- $R_T$ supercommutes tith the $C^{p+1,q+1}$-structure on $\hat{H}$.
- The pair of subspaces $(\Gamma_0,\Gamma_T)$ is a Fredholm pair.
- The subspace $\Gamma_T$ does not intersect $\Gamma_\infty$.
$\newcommand{\eFL}{\mathscr{FL}}$. We denote by $\eFL^{p+1,q+1}(\hat{H})$ the set of closed subspaces $L$ of $\hat{H}$ such that the reflection $R_L$ in $L$ satisfies the conditions $1$ and $2$ above. (In the old paper I called these spaces generalized lagrangian spaces of type $(p+1,q+1)$. We have thus produced an inclusion
$$\eF^{p,q}(H)\to \eFL^{p+1,q+1}(\hat{H}). $$
One can show two things. First, the space $\eFL^{p+1,q+1}$ classifies $K^{p+1,q+1}$ and second, the above inclusion is a homotopy equivalence. (The proof uses Bott periodicity.) One can use a process of symplectic reduction to canonically associate to continuous family $L: X\to \eFL^{p+1,q+1}(X)$ an element
$$ (E,\eta_0,\eta_1)\in K^{p+1,q+1}(X)\cong K^{p,q}(X). $$
Observe that the elements of $\eFL^{p+1,q+1}$ can be identified with selfajoint operators $R:\hat{H}\to \hat{H}$ such that $R^2=1$ and super-commute with the $C^{p+1,q+1}$-structure and they satisfy condition 2. This almost looks like a Kasparov element.
To actually get a Kasparov element consider a smooth, odd, nondecreasing function $\newcommand{\bR}{\mathbb{R}}$
$$\beta : \bR\to \bR,\;\;\beta (t)=\pm 1\;\;\mbox{if $\pm t>1$}. $$
For $\newcommand{\ve}{{\varepsilon}}$ $\ve>0$ we set $\beta_\ve(t)=\beta(t/\ve)$.
Then for any $T\in\eF^{p,q}$, $\beta_\ve (F)$ defines a Kasparov element for $\ve>0$ sufficiently small.
Remark. I will describe below an explicit map
$$K^{p,q}(X)\to [X, \eF^{p+1,q+1}]. $$
This is more or less what you need since $K^{p,q}(X)\cong K^{p+1,q+1}(X)$ and more generally, $K^{p_0,q_0}(X)\cong K^{p_1,q_1}(X)$ if $p_0-q_0\equiv p_1-q_1\bmod 8$.
The symplectic point of view is crucial since the above map is inspired by Floer's work on symplectic (Floer) homology. To justify the symplectic terminology let me discuss a simple example.
Let us look at a $C^{1,0}$-module. This is a real Hilbert space equipped with a orthogonal operator $J: H\to H$ such that $J^2=-1$, $J^*=-J$. In the finite dimensional case think $H=\bR^n\oplus \bR^n$, $J(x,y)=-(y,x)$.) A subspace $L\subset H$ is called Lagrangian if $JL=L^\perp$.
In the case of $\bR^n\oplus \bR^n= T^*\bR^n$ equipped with the canonical symplectic structure we obtain in this fashion the classical notion of Lagrangian subspace. If $P_L$ denotes the orthogonal projection onto $L$ and $R_L=2P_L-1$ denotes the orthogonal reflection in the subspace $L$, then $L$ is Lagragian iff $R_L$ anticommutes with $J$,
$$R_LJ+JR_L=0. $$
In algebraic terms $R_L$ defines a $\newcommand{\bZ}{\mathbb{Z}}$ $\bZ/2$-grading of the $C^{1,0}$-module $H$. However the symplectic point of view has more flexibility because it leads to certain operations which algebraically do not seem natural. (I'm thinking here of the process symplectic reduction.)
In general given a $C^{p,q}$-module $H$, we define a $(p,q)$-Lagrangian in $H$ to be a subspace $L\subset H$ such that $R_L$ supercommutes with the $C^{p,q}$-structure, i.e., $R_L$ is a $\bZ/2$-grading of the $C^{p,q}$-module $H$. Denote by $\DeclareMathOperator{\Lag}{Lag}$ $\Lag^{p,q}(H)$ the space of $(p,q)$-lagragians in $H$. Observe that and element in $K^{p,q}(X)$ is defined by a continuous map
$$ X\to {\Lag}^{p,q}(H)\times {\Lag}^{p,q}(H), \;\;x\mapsto (L_0(x), L_1(x))$$
where $H$ is a finite dimensional $C^{p,q}$-module. Thus an element in $K^{p,q}$ is a pair of continuous families of lagrangian subspaces in a $C^{p,q}$-module. Moreover, it suffices consider only the case when one of the families $L_0(x)$ is constant.
Fix a finite dimensional $C^{p',q'}$-module $H$, $p'=p+1$, $q'=q+1$. Denote by $J$ the operator on $H$ defined by the multiplication by $e_{p+1}$ so that $J^*=-J$, $J^2 =-1$.
To a pair of lagrangians $L_0, L_1\in \Lag^{p',q'}(H)$ we associate an operator $T_{L_0,L_1}\in\eF^{p,q}(\bsH)$ where
$$\bsH=L^2(0,1, H), $$
and $T_{L_0,L_1}$ is the closed, Fredholm selfadjoint unbounded operator on $\bsH$ with domain
$$ D(T_{L_0,L_1})=\bigl\lbrace u\in L^{1,2}(0,1; H);\; u(0)\in L_0,\;\;u(1)\in L_1\bigr\rbrace, $$
such that
$$ T_{L_0,L_1} u(s)=J\frac{du}{ds},\;\;s\in (0,1). $$
Above, $L^{1,2}$ denotes the Sobolev spaces of functions with first order derivative in $L^2$.
The motivation for the operator $T_{L_0,L_1}$ comes from symplectic Floer homology. In his papers on the (Floer) homology of a pair of lagrangian submanifolds A. Floer investigated the operators $T_{L_0,L_1}$ in the case $p=1, q=0$ and the indices of one-parameter families of such operators. Note that only the domain of $T_{L_0,L_1}$ depends on $L_0,L_1$. The action of $_{L_0,L_1}$ is independent of the lagrangians $L_0,L_1$>
To an element $\alpha\in K^{p',q'}(X)$ represented by a continuous map
$$ X\ni x \to (L_0(x), L_1(x))\in {\Lag}^{p',q'}(H)\times {\Lag}^{p',q'}(H) $$
we can associate an element $T_\alpha\in [X,\eF^{p,q}]$ given by the continuous map
$$ X\ni x\mapsto T_{L_0(x), L_1(x)}\in\eF^{p,q}(\bsH). $$
As mentioned before, $\eF^{p,q}(\bsH)$ classifies $K^{p,q}$ and thus the map $T_\alpha$ defines an element ${\rm ind}\; T_\alpha\in K^{p,q}(X)$. In that old paper I proved that ${\rm ind}\; T_\alpha \in K^{p,q}(X)$ coincides with $\alpha\in K^{p',q'}(X)$ via the canonical isomorphism $K^{p,q}(X)\to K^{p',q'}(X)$. The proof uses crucially the process of symplectic reduction. For details see Thm. 5.5 in the old paper.
The familly $ T_{\alpha}$ can be given a Kasparov descriptition as explained above.
