Ceramic capacitors microphonic bandwidth
The microphonic bandwidth is not decoupled from the electrical bandwidth of the capacitor.  The piezoelectric effect will cause strain in response to an electric field, or generate an electric field in response to strain.  If you can use the capacitor at that frequency, then it is vibrating at that frequency, and will happily generate voltage noise if vibrated at those frequencies.  In fact, it will generate noise for itself due to local mechanical ringing.  

In other words, there is no free lunch.  The microphonic bandwidth can be assumed to be close to the AC bandwidth of the capacitor.  If you're sampling at 200kHz, and expect a few ppm (a few µV) or less of noise, then, to put it bluntly, you cannot use an X5R or X7R ceramic capacitor.  

Note, many film capacitors will also exhibit a degree of microphonics due to having a soft, flexible dielectric and vibration causing compression waves through the plates.  But they generally will be much lower - but still possibly in the microvolt range. 

I am curious why you have seemingly already ruled out the best candidate, a nice little 10µF tantalum capacitor.  Sure, there are doubtless people groaning about their high leakage current right now, but that is silly.  Who cares?  You're storing charge for 5µs.

The first hit for low leakage tantalums yields this datasheet for a nice line of capacitors by Kemet.  A quick scan shows me they have a 10µF Tantalum capacitor with 800mΩ ESR and DC leakage of 1.5µA.  That translates into a voltage change of 160nV over 5µs.  In other words, who the hell cares?  If you can tolerate higher ESR, there is one that will change only 80nV.  And there are plenty of other tantalums that have much lower ESR but also have leakage that would place you below a 1µV of change for 5µs.  

The thing is, tantalums are naturally very low noise, and based on the information you've given, I would think they would be the ideal choice for a sample and hold application considering how very brief the 'hold' time is in your circuit.  

If you still can't use tantalums for whatever reason, then I strongly recommend Polyphenylene sulfide film capacitors, as these have voltage noise almost as low as C0G.  Unfortunately, a 10µF PPS capacitor will be very chub indeed, the ones on mouser look to be $15.  But no one said precision was cheap!