Extending the bandwidth by 1.5 or 2 is quite feasible depending on how tight the actual filtering is (for example with a simple first order roll-off, a 6 dB peaking properly placed will extend the bandwidth by two). I provide three different approaches for creating FIR filters to pre-emphasize for passband droop in bandwidth restricted channels: Using a least-squares equalizer when the stimulus can be modified to be either a sounding chirp or pseudo-random sequence, or determining the channel from the FFT of the OP's stimulus and response, or using a 3 tap peaking filter commonly used as an inverse Sinc compensator. Each are described in more detail below with links to other posts providing further details and example code.
My recommended approach in the case where optional sounding waveforms can be used would be a least squares approach using the "Wiener-Hopf" equations. I provide further details on how this works and have example MATLAB code that determines either the transfer function or the equalization for the transfer function (by swapping tx and rx in the function) at this post.
The use of this function is also demonstrated here showing how the delay for the channel can also be determined from the derived channel response.
If this approach is to be used, and assuming the stimulus could be modified, it is preferred to use a stimulus that is spectrally rich, as the frequency response can only be determined at frequencies where energy exists (and the error in the result is inversely proportional to the SNR received at any given frequency). For this reason, useful stimulus patterns are either frequency chirps or pseudo-random (PRN) sequences, which can be implemented with linear feedback shift registers. An example implementation of a PRN including MATLAB code is at this post. (That one is a GPS C/A Code generator as the sum of two PRN's but demonstrates how simple a single PRN can be constructed).
Alternatively the frequency response can be estimated from the FFT of this input and output of the OP's waveform, such that the input FFT is divided by the output FFT, and then the inverse FFT of that result can be the compensation filter. The result of this in the frequency domain should be reviewed first due to the likelihood of noise enhancement at out of band frequencies that should be reduced prior to the inverse FFT. This won't be as accurate as the recommended approach above, but may be sufficient and easy to generate.
Finally a third and very simple approach for high frequency enhancement is this 3-tap linear phase inverse Sinc filter which is general is parametrized for various values of high frequency peaking at the band edge of the waveform. This is used as pre-emphasis for the pass-band droop introduced by D/A converters and CIC filters, so may be sufficient for enhancing the bandwidth in the OP's case.
