Frequency dependent frequency shift
Before I try to answer the question, let's see how the question relates to dispersion.

The solution space of a dispersive (linear) wave equation in $n$ dimensions is spanned by the basis functions

$$f_{\mathbf{k}}(\mathbf{x},t) = \exp( i\, \omega(\mathbf{k}) t )\exp( i\, \mathbf{k}\cdot \mathbf{x} )$$

where $\mathbf{k} \in \mathbb{R}^n$ is the spatial frequency vector or wave vector. The temporal frequency $\omega$ depends on the  wave vector, or just on its magnitude $|\mathbf{k}|$ if the dispersion is isotropic. If $\omega \propto |\mathbf{k}|$ we have the special case of no dispersion. We can also assume in general that $\omega\geq 0$ to simplify calculations.

The basis functions are orthonormal in the Dirac sense with respect to their spatial coordinate:

$$\langle f_{\mathbf{k}},f_{\mathbf{k'}}\rangle := \int_{\mathbb{R}^n} f_{\mathbf{k}}(\mathbf{x},t)\, f_{\mathbf{k'}}(\mathbf{x},t)\, d\mathbf{x} = \delta(\mathbf{k}-\mathbf{k'})$$

And the general solution of the wave equation is:

$$f(\mathbf{x},t) = \int_{\mathbb{R}^n} c(\mathbf{k})\, f_{\mathbf{k}}(\mathbf{x},t)\, d\mathbf{k} $$

If we now observe that general solution at $\mathbf{x}=0$, we get the relationship

$$f(0,t) = \Re\left(\int_{\mathbb{R}^n} c(\mathbf{k})\,\exp( i\, \omega(\mathbf{k}) t )\, d\mathbf{k}^n \right)$$

which looks similar to a Fourier transform of $c(\mathbf{k})$. If we can turn this into a proper Fourier transform, the inverse Fourier transform of the observation will give us the expansion coefficients $c(\mathbf{k})$ required for the reconstruction of the full spatial solution of the wave.

At this point, we need to make a few additional assumptions. The first one is that $n$ equals 1 so that the coefficients are also a function on $\mathbb{R}$. We also have to assume that the dispersion is isotropic and $\omega(|\mathbf{k}|)$ is differentiable and strictly monotonic.

This is not enough however. We also need to assume a direction of travel so that $\omega(|\mathbf{k}|)$ becomes an invertible function on a restriction of $k&gt;0$ or $k&lt;0$ respectively.

We pick a "right" travelling wave, so $k&gt;0$. With this, we can change the integration variable with $dk=\frac{\partial k}{\partial\omega}\,d\omega$ to get

$$f(0,t) = \Re\left(\int_\mathbb{R_{\geq 0}} c(k(\omega)) \frac{\partial k(\omega)}{\partial\omega} \exp(i\,\omega t)\, d\omega \right)$$

where $k(\omega)$ is the inverse function of $\omega(k)$. Identifying 

$$C(\omega) := c(k(\omega)) \frac{\partial k(\omega)}{\partial\omega}$$ for $\omega\geq 0$ and the complex conjugate for $\omega&lt;0$, we can see that $f(0,t)$ is the Fourier transform of $C(\omega)$.

With the inverse Fourier transform we get

$$C(\omega) = \int_\mathbb{R}f(0,t) \exp(-i\,\omega t) dt$$

and consequentially

$$c(k(\omega)) = \frac{1}{k'(\omega)} \int_\mathbb{R}f(0,t) \exp(-i\,\omega t) dt$$

or simply

$$c(k) = \omega'(k) \int_\mathbb{R}f(0,t) \exp(-i\,\omega(k) t) dt$$

which we can express in terms of the inverse Fourier transform as

$$c(k) = \omega'(k)\,\mathcal{F_t}^{-1} \left\{ f(0,t) \right\}\left( \omega(k) \right) $$

Using this theoretical background we can now reconstruct the spatial wave function of a free one-dimensional right travelling wave with open boundary conditions.

In the expansion of the general wave solution from above we can substitute the $c(k)$ to get:

$$f(x,t)=\int_\mathbb{R} \omega'(k)\,\mathcal{F_t}^{-1} \left\{ f(0,t) \right\}\left( \omega(k) \right)\, \exp( i\, \omega(k) t )\exp( i\, k x ) dk $$

This can also be written as a Fourier transform in $k$:

$$f(x,t)= \mathcal{F_k}\left\{ \mathcal{F_t}^{-1} \left\{ f(0,t) \right\}\left( \omega(k) \right)\,\omega'(k)\,\exp(i\,\omega(k)\,t)\, \right\}(x)$$

This translates to the following algorithm for the reconstruction of the spatial wave function at time $t$:


Take the inverse Fourier transform of $f(0,t)$
Evaluate the result at $\omega(k)$
Multiply the result with $\omega'(k)$
Evolve the result in time by multiplying with $\exp(i\,\omega(k)\,t)$
Take the forward Fourier transform with respect to k


You can implement this in the discrete domain by taking regular grids for the variables $\omega$ and $k$ and use the discrete Fourier transform.

As you can see, it's a little more complicated than what you described, but your idea was going in the right direction.