Anywhere where the two terminals of the battery are connected, wherever it's most convenient. If there's a wire soldered to the circuit board going to the battery, the solder point is fine. If the battery terminal is exposed rather than insulated, also fine. Any solder pad on the circuit board connected to a terminal, also. Use a multimeter set to the appropriate voltage range. It's possible that you get a measurable voltage across the charging contacts on the transmitter related to the battery voltage; if you hadn't already opened it I'd have suggested probing the contacts immediately after the 30-second-lasting charge to see if there's a voltage that decreases rapidly. Avoid bridging or short-circuiting any connections when applying the probes.
Whether you try to rejuvenate or test, or simply replace the battery, you'll have to disconnect it. For a battery connected by wires rather than pins through the board, cutting them is often the best way. It also makes replacing easier (if you don't cut the wires too close to the board), unless the assembly is so tiny that's there's no room to pack in soldered insulated joints.
Regarding the charger current rating: it just has to be capable of delivering the required charging current, higher rating is fine but ignored. I've found the Zeo battery seems to indicate full charge in about 1 hour (if memory serves, I did check it long ago), so the actual charging current is probably about 50mA (it would be, say, 5mA for a 10-hour charge, and so on). Any charger that can supply AT LEAST that current is fine. The manufacturer would buy a standard charger; 0.5A at 5V is simply probably the cheapest these days. If you want to fill tiny medicine bottles with water, you wouldn't install a special tap that delivers a trickle, you'd just use an ordinary tap turned on just enough to deliver a trickle.