Never owned a similar device, but judging from the product pictures:
Indeed, the charging logic for the battery is not inside the handset, but the charging cradle, and hence, that would quite likely be confused should you transmit while charging, as that looks like a sudden drop in battery voltage.
So, the other option is what Hytera sells as "Battery Eliminator", which has the shape and handset contacts that a battery has, but is actually nothing but a step-down voltage converter giving you something in the range of 7.4 V, plus probably just a resistor to emulate the temperature sensor of the battery pack.
If it's that easy, you can either buy the Hytera battery eliminator, and attach the cigarrette lighter plug to a 12V supply, or you can zurechtbastel something that just happens to have contacts in the right places:
Check the number of contacts on the device side of the battery.
- 2 only: simplest case, you just need to connect a 7.4 V power supply to these (watch polarity)
- 3 or 4: there's a sensor to measure the battery temperature built in to the battery pack, and your handheld might check that to make sure the battery isn't overheating.
In that case: you need to emulate that sensor:
Verify which pins are the positive and negative contact with a multimeter.
Measure whether there's a voltage between the other two (4 contacts), or between the third and the negative contact. If that's the case, it's a thermocouple. If no reasonable voltage is found:
put the battery in the fridge, measure the resistance between the two contacts that aren't + and -. Let it warm up (sun, heating, body warmth!), and compare. If it drastically changes, yep, there's a thermistor attached here, and you might need to emulate that so that the handset doesn't think the battery is overheating.