The theory of Fourier transform for Euclidean spaces has analogues for locally compact abelian groups. In the noncommutative setting, representations can be used to define analogous transforms. My question is whether the following naive approach to defining a local transform for Lie groups has any of the interesting properties of the usual FT. (I have never seen this definition in the literature and suspect it might be due to its lack of maintaining the usual properties of FT).
Since $\exp\colon \mathfrak{g}\to G$ is a local diffeomorphism from a neighborhood $U$ of $0$ to a neighborhood $V$ of the identity, define for $\exp X=x\in V$ $$ \hat{f}(x):=\mathcal{F}(\chi_U\cdot f\circ \exp)(X) $$ where $\chi_U$ is the characteristic function of $U$, and hence the right-hand-side is the Euclidean FT of the restriction of $f\circ\exp$ to $U$.
This is not an answer, but it's too long for a comment.
Classical Fourier transform can be seen as an isomorphism between $L^2$ functions on $S^1$ or $\mathbb{R}^n$ and $l^2$ sequences given by a particular choice of basis functions. These functions are actually group characters. Now group characters encode the irreducible representations of these groups.
So the usual generalizations of Fourier transform investigate irreducible (sub)representations inside $L^2(G)$. While it is true, that this study can be to a large extent reduced to study of $\mathfrak{g}$-representations, your definition seems rather arbitrary. You should at least deal with dependency on $U$. For example by taking direct or inverse limit.