MCP73871  MCU interface
the pull-ups would power the charger IC from battery via this diode
  when no USB power cord is present.


The charger IC is already "powered" by either the input voltage or the battery voltage, internally, as the data sheet explains:

The MCP73871 device automatically obtains power for
the system load from a single-cell Li-Ion battery or an
input power source



So, connecting pull-up resistors to either of these voltage inputs or the output is absolutely fine.