I asked Playbonk about this.

He stated that, to his knowledge, he can't make the charging handle move without Blt, since the vanilla M4A1 anim does not involve the charging handle.

@RJC9000 maybe mw2019 animations for car-4 then? is doesnt involve charging handle iirc

You are free to send Playbonk $10 for that animation comission for MW2019 AR15 anims, or $200 at splish for a full set of custom anims.
:P

I wasn't involved in the animation work, I just threw money at Playbonk and told him that I wanted a recreation of the XM4's empty reload.

As for that MW2019 animation style that involves swapping out the old magazine while holding the new magazine, I am not sure it is possible with vanilla weapons due to most vanilla weapons only having one magazine object.
You'd have to ask Playbonk for more details.