Is it possible to in some way change this number, and thus change the identification of the plane to something innocuous?
Short answer. NO
Modern transponders use what is called Mode S transponders. Since the mid nineties. Mode S transponders are more accurate and do alot other things not worth mentioning. Most importantly Mode S contain the unique aircraft identification.
Long answer. Yes, an aircraft id can be changed but not during flight and while transmitting.
The following from en.wikepedia.com
Upon interrogation, Mode S transponders transmit information about the aircraft to the SSR system, to TCAS receivers on board aircraft and to the ADS-B SSR system. This information includes the call sign of the aircraft and/or the transponder's permanent ICAO 24-bit address in the form of a hex code.
Mode S equipment on aircraft are assigned a unique ICAO 24-bit address or (informally) Mode-S "hex code" upon national registration and this address becomes a part of the aircraft's Certificate of Registration. Normally, the address is never changed, however, the transponders are reprogrammable and, occasionally, are moved from one aircraft to another (presumably for operational or cost purposes), either by maintenance or by changing the appropriate entry in the aircraft's Flight management system.
Thank you for taking the trouble to write out that lengthy and very clear explanation. I appreciate it.