Reconstructing/interpolating small regions of a bandlimited signal by taking the fewest possible samples
I have some very short signals in the range of 8 to 16 samples. These represent a bandlimited signal, sampled at or slightly above the Nyquist rate. 


Nope. A signal can't be limited in time and in frequency at the same time. If it's very short, than chances are the bandwidth is a lot higher than you think it is and that you've already picked up some significant aliasing in the sampling.


  I've experimented with sinc interpolation with a few different windowing methods, and also tried numerically optimising a kernel to achieve the interpolation, without success.


Can you elaborate what your problem is? If it's sampled correctly, sinc interpolation typically does a good job, although it does destroy any causality and will extent the signal in time. But that goes back to my first point: a band-limited signal can not be limited in time as well, so it can't be causal either.