External power connector wiring varies greatly from one plane to next. On some you can charge the battery through it; on others (like mine TB20) a power connection there operates a relay which disconnects the battery... but some early TB20s can be charged via the external power connector.
The US made Battery Minder aviation battery charger stands out above the rest in the tests. I got a quote for the Euro version a few months ago: E24041-AA-S2 $197.00 USD plus shipping.