Would I be correct in thinking that not having the latest firmware matters more with respect to RTK than static/post-processing? I've never researched this, so I may be wrong.
Not necessarily. I can recall one firmware update to the Ashtech ProMark2 that was required to allow the receiver to function reliably (and in some cases, at all) after the launch and activation of an EGNOS satellite. Another firmware issue resulted in static files collected with Trimble 5800 receivers to cause PAGES (the OPUS baseline processor) to choke. So it's not just RTK stuff.
We recently upgraded out Topcon GPS firmware and have been surprisingly impressed with better reliability of the Glonas sats in RTK. I was constantly turning off the Russians SV#46-51 to get fixed and now they seem to work great.
I reccommend Firmware upgrade.
John