Improving spectral response with lost downsampled data
must maintain a specific FFT length and small FFT bin width, down at baseband.


Well, the FFT bin width is defined as $\frac{f_\text{sample}}{N}$, with $N$ being the FFT length and $f_\text{sample}$ the sampling rate of the signal that undergoes the FFT. So, if both these parameters are fixed, you have no freedom in choosing the bin width whatsoever.

Note that this describes a pure view on the "tool" FFT. My guess is that you want to build some spectrum estimation – and you can certainly build a "finer" estimator.


  Is there a way to use the lost samples to improve the spectral response of the FFT and/or the SNR, given the constraints?


Since if done without aliasing, the downsampling removed no information from the signal you're interested, no.

What you can, however, do is build a better frequency estimator; that might be using the same FFT length multiple times (or, in fact, even build a larger FFT from your small FFT used multiple times), but it's hard to advise here: your resource constraint doesn't give us anything to go on – and FFTs are typically highly optimized, so if a larger FFT itself won't fit/run on your system, then chances are that complex estimators won't, either.