From appearances your setup should work. You are giving the emitter a direct path to V+. You are biasing the PNP to ground. This gives maximum sensitivity. The bias can be as small as 0.6 V E-B and you should get flow into the battery.
A battery does odd things in a circuit. It can cause unexpected volt levels to appear at nodes. It can't be simulated by a capacitor because a capacitor has very high resistance unlike a battery.
However what is your charging device? You say it's current limited. Is it the kind that shuts off unless it detects a sufficiently low load resistance, and then it turns on?
The 600 ohm bias resistor is the only load, at the start. Is that able to turn on the charging device?