Charging 1S 850 mah battery at low current, 4.2 volts max, potential issues
If it is really true that the battery charging circuit never puts out more than 4.20V then the Lipo should be OK. However you must also ensure that it doesn't go below 2.8 V or it could be damaged by over-discharge, and attempting to recharge it might blow it up! If the Lipo doesn't have a built-in PCM (Protection Circuit Module) then you should add one.

Another option might be to simply replace the 3 x AA NiMH pack with LSD (Low Self Discharge) cells, eg. Eneloops. This will provide much longer backup time with greater safety.


  When running on battery power, it is reported the board pulls about
  1.3 ma current.


At 1.3 mA an 850 mAh Lipo would last about 3 weeks. AA LSD cells should last about 8 weeks. If you don't use the computer regularly then you might want to put a switch in the battery circuit to turn it off for long-term storage.