Excuse me....
Its sounded like you were recharging the 1/2 AA lithium. Could be me. Its 5am and been working an all nighter.
If the single lithium battery does not glow or is very dim the 1/2 AA battery even at 3.0 volts is not producing enough to run the charge side. Could run the transmitter side but not enough for the critical side.
From looking at the Everready site, which ever of the two they list, it look like burst current for a few seconds and its spect to drop to 2v at 800ma which may be barelly enough to do the job.
Use Alk battries. If you substitue for them you need to match voltage levels and current draw.
2 std size AA Lithium can do it but not one of these small one cell 3v ones. They have a temp current limited built in.
PS: If that the same battery in your picture, that looks like one you would use to power the clock/cmos in a computer. Those are designed for low sustain current draws while this is a high current draw application. The circuity could go to 5v max of the TX-2 chip. The circuity is designed for 3v and drops when charging.