If your diode allows 0.2 mA when reversed biased from 5V to GND then it basically looks like a 5V source in series with a 25K resistor. Which would explain why it turns Q4 on in your circuit.
This is a pretty high impedance source. When the battery is present it will look like a voltage source with a much lower impedance. You might be able to exploit this.
One way is to replace R33 with a short and change R35 to something like 2.2K. That way, when the battery is absent, the bulk of the voltage drops across the diode instead of R35. This will prevent Q4 from turning on. When the battery is in the circuit then the diode isn't really in play since it will be forward biased so the battery's voltage will be dropped across R35--turning Q4 on.
I'd be a little worried about the BAT pin latching Q4 on, however. Once it turns on, this scheme wouldn't be able to determine whether or not a battery is present. I don't know enough about the charging IC to know if this would be a problem or not.