I know that, you asked if we'd ever seen transporters in that era perform that task. And we did, and on top of that the Crossfield class has be shown to have better transporters than the Enterprise and a unique targetting aparatus that operates over extreme distance that Tilly was specifically assigned to the ship to help manage.
So it's possible for ordinary transporters to do it (and across dimensions etc etc in TOS), the ship has a long range targetting system, multiple reactors to help power such systems and the task was performed by someone with a specialised skill in doing just that kind of thing.
So no problems there.
That's cleverly argued, but it seems simpler to go with Cyrus's suggestion that there was just a time cut while the ship traveled to her planet.