Clyde,
the type of charger I have is of the type you described.
The problem is what you said: if the battery charger recharges the battery, there is no end-of-charge confirmation.
I think the charging circuit, present in the after-market cover, actually blocks the power supply.
For this the original charger LED turns green.
But, if I use the RC charger (configured unbalanced), what happens when the charging circuit, contained in the cover, cuts the power supply?
Does my RC charger continue to pump energy or does it notice that it no longer requests power?
It seems to me that you did this test.
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73 by Max IK0TIX