I apologize for the vagueness of the question, but I'm working more from high-level intuition here than from rigorous formalism. In short, my question is this: although the Laplace "basis" is not orthogonal, does there exist a unique transformation from the time-domain to the Laplace domain that is properly interpretable as a change-of-basis?
The Fourier transform can be viewed as a "change of basis" of a function space from a delta-function basis to a sine-wave basis, accomplished by the taking of orthogonal projections.
The Laplace transform works similarly, except the Laplace "basis" is not orthogonal. This presents us with a slight problem, as a single-frequency input to the Laplace transform does not yield a delta function output (as it would for a Fourier transform) - instead, you get a "pole", whose value blows up like $\frac{1}{s-a}$. Accordingly, this complicates the inverse Laplace transform c.f. the inverse Fourier transform; the latter can be thought of as simply "adding up" the orthogonal components to reconstruct the original function, while the former clearly cannot (as the sum of the projections would "over-count" for any present frequency if they were naively added up in such a way).
This means that the Laplace transform is not truly interpretable as a simple change-of-basis. But is there any alternative transform that is interpretable in such a way?
In finite-dimensional spaces, non-orthogonality of a basis is not fatal to finding a unique transformation into the coordinates of that basis (to wit, we can simply invert the column-matrix of the basis vectors). Is there a similar trick that can be done for infinite dimensional spaces that could be used here?
Stating that "The Fourier transform can be viewed as a "change of basis" of a function space from a delta-function basis to a sine-wave basis, accomplished by the taking of orthogonal projections" is vague and incorrect. One has first to decide in which space the Fourier transform operator $F$ is supposed to act. If it has to act on Dirac distributions, then supposedly this space should be the space $\mathcal{D}'$ of distributions or $\mathcal{S}'$ of tempered distributions. But these spaces are not equipped with inner products, and their topology cannot come from an inner product. Therefore mentioning orthogonal bases makes no sense, with one exception, that is the discrete setup of the Hilbert space $L^2(\bf{Z})$. In this setup the Fourier transform is just the Fourier series, and it is true that the Dirac masses $\delta_n$ and $\delta_m$ are orthogonal if $n$ is different from $m$, and so are their images under $F$, namely $e^{imt}$ and $e^{int}$. In this setup the "Laplace transform" is just the Zeta function operator, that is, the natural complexification of the Fourier series, but it needs to be truncated to positive half planes, that is, to be regarded as a Laurent series, otherwise the exponentials $e^{inz}$ are unbounded. In which inner product should we look at orthogonality? Now these exponentials are functions on the complex plane, not on the unit circle, and they are unbounded. Instead, if one looks at $L^2(R)$ instead than $\mathcal{S}'(\bf{R})$, all this makes no sense: the Dirac measures do not form an orthogonal basis. A basis is a collection of functions in $L^2$ such that all other function in $L^2$ is a finite linear combination of them. But the Dirac measures are not in $L^2$ (and neither are the exponentials $e^{ixt}$), and the Fourier inversion theorem states that all functions in $L^2(\bf{R})$ can be obtained not by taking finite linear combinations of these exponentials, but instead integrals with $L^2$ weights.