Can an Eldritch Knight use their Arcane Charge feature before falling?

Question

This Q&A establishes that if you suddenly find yourself mid-air (e.g. by teleportation), you only have time to use a reaction before starting to fall (examples include casting feather fall, or using a readied action if you previously took the Ready action set the trigger to something like "when I am teleported").

This question is a follow-up to that one specifically concerning the Eldritch Knight's 15th-level feature, Arcane Charge:

At 15th level, you gain the ability to teleport up to 30 feet to an unoccupied space you can see when you use your Action Surge. You can teleport before or after the additional action.

For reference, the Fighter's Action Surge feature says:

Starting at 2nd level, you can push yourself beyond your normal limits for a moment. On your turn, you can take one additional action on top of your regular action and a possible bonus action.

Now, the question:

On their turn, a 15th level Eldritch Knight casts misty step to teleport 30 feet straight up. Because Action Surge is a "free action", can they activate their Action Surge and use their Arcane Charge to teleport another 30 feet upwards before starting to fall?

Why the answers to the other Q&A do not answer this question:

• Using an Action Surge is not a reaction (thus it's not clearly possible to do it before falling, according to the other Q&A)
• Using an Action Surge is not an action or bonus action (thus it's not clearly impossible to do before falling, according to the other Q&A)
• In fact, an Action Surge is a "free action" (although no such game term actually exists in 5th Edition) whose only timing restriction is that it can only be done on your turn.

Answer 1

I would rule YES

Here's my reasoning: As per the Q&A, you have enough time to perform a reaction before falling, but Action Surge is instantaneous. Therefore, from my reading, and from my love of "Rule of Cool" rulings, it's possible.

However, There's an argument to be made against this ruling, which I'll include for the sake of completeness: Arcane charge states "You can teleport before or after the additional action", which could be interpreted as the teleport being part of the action, in which case you do not have time to do so before falling. Personally, I think this line of reasoning is pretty weak, which is why I as a DM would allow it (teleporting before falling).