The normal meter is moving-coil (+fixed magnet). It is polarised, because of the permanent magnet. It is very sensitive (50uA), fast acting, and these are usually what you want.
There is a type of meter called a moving-iron (+fixed coil) meter. The iron is attracted into the coil. There is no permanent magnet. and so it is not polarised. South pole attracts iron just the same as North pole.
They will read the avarage of AC or DC just the same.
You can use a moving iron meter in the AC or DC lines.
Because the big lump of moving iron is much heavier than the teensy self-supporting coil of a moving coil meter, moving iron meters are much slower acting. This would be bad for sound level, but is excellent for AC / dirty DC, where you don't want the needle buzzing around at 100Hz. Also you don't get much magnetic force from a few turns of wire and a lump of iron, compared to using a strong magnet. This makes moving iron meters not very sensitive. But again, this is just what you want for a 5A meter. The meter can be made without a current shunt, using a few turns of thick wire - cheaper and more robust.
Note that in your circuit the meter is only seeing DC and half the current. A moving iron meter could be used in any of these positions.
simulate this circuit – Schematic created using CircuitLab
Simple battery chargers are current limited by the leakage inductance of the transformer, shown separately as L1. If you look closely at the transformer, you will see that the laminations are not stacked normally EIEIEI but something like EEEIIIEEEIII this makes the magnetic coupling worse, incraesing this leakage reactance and making the transformer saggy . Toroidal transformer are the exact opposite - fantastic magnetic coupling,little sagginess.