What strain rates are you looking at? Do they have to be so high? Even typical blast loaded composite panels won't experience the sort of strain rates an SHB typically applies.

SHB is very high rate and I suspect you'll have difficulties in determining elastic props with non-linear deformation due to rates in effect. Strain gauges will effectively average strain over an area so if the deformation gradient is not constant then you will just get smeared out properties over the area you're measuring, assuming your signal conditioning and data logging rates can come anywhere close to what would be needed for a SHB. Drop towers often have piezoelectric load cells that would allow you to measure load history at sufficient frequencies (assuming your data logging kit is up to it) and optical (e.g. laser or high speed camera based) methods would allow you to determine displacement/strain. High speed cameras can be particularly useful for high rate testing; my group use them for Digital Image Correlation for impact, high rate and blast testing and it works well.

If you can see the sample in the impact area then you may be able to use high speed cameras and DIC to visualize the strain field which you might be able to use to calibrate a model to determine elastic props . This would be kind of analogous to backing out traction-separation relationships for cohesive zones through FEA (don't have to model with commercial/conventional FEA, or even FE to solve the governing equations. might be faster/easier to do with something like Matlab ).However, I expect this would be time consuming and would probably be pretty tricky as you'd need apriori knowledge of the general relationship between rate and modulus and if you're using a SHB, I expect you'll get damage at which point backing out moduli would get really tricky/impossible!

Alternatively, and the easiest, you may be able to establish a rate dependency for lower rates, and at various fibre orientations, with a DMTA machine and then go from there.