Closing the loop, yesterday I soldered the ESP8266 parts to a PCB (the previous version looked like a Dupont wire orgy and also used a different 3.3v voltage regulator), and then ran RSSI measurements using a narrower bandwidth and my now more evolved methods for capturing RSSI noise.
The good news is that with a regular Moteino R4 with a wire antenna, and with the improved ESP8266 just a couple inches from the antenna, there was no meaningful difference in the amount of RSSI noise detected.
For those interested in the details of the comparison, see:
http://pastebin.com/vC5bx0fsand
http://pastebin.com/g2tqNi3W[Edit1: the above two measurements were taken with the Moteino R4 and ESP8266 (an ESP13) on separate circuits, each with their own power supplies run from batteries. The good news is that powering them both from the ESP8266's power supply (an LM1117T-3.3v powered by four AA batteries), yields the same level of RSSI noise as when the R4 and ESP13 were not directly connected:
http://pastebin.com/sCmXAhwg]
[Edit2: OK, I just connected the Rx and Tx of the ESP13 to the Moteino R4, remeasured, and also no meaningful change in RSSI noise: average RSSI noise measured over 1000 samples at this same narrow bandwidth is -118.79. BTW, since all this is just an experiment, I am using Dupont wires to connect the ESP13 to the R4. ]