Got a chance to look at where the probe's high noise floor is coming from. Turns out I don't have to make my own spreadsheet, someone's already made a noise calculator. http://www.jensign.com/noise/ When I plug in the parameters for the OPA847 and surrounding parts, I get 636uVrms of noise. (3900MHz gain bandwidth, 0.85nV/Hz voltage noise, 2.5pa/Hz current noise, 30Kohm Rf, Cf around 0.2pf, Cin around 7pf) That's consistent with my 7-10mV peak to peak noise I'm seeing after the 3-6x gain from the LT1819. (a quick rule of thumb is that peak to peak noise is about 6x RMS noise)
This calculator also shows the individual contributions to the amplifier noise. In this case, Op-amp input voltage and current noise dominates. A noise-free op-amp would drop my noise floor about 5x. Looks like it's worth spending time to develop a better amplifier. It also looks like I should model the design first in LTSpice as Google show's it supports noise analysis in addition to transient and frequency response analysis.