What a great question.

>> Who am I speaking to?
(This is grammatically dodgy.)

It should be:
>> Whom am I speaking to?
(But, this is a bit pretentious.)

>> To whom am I speaking?
(This is grammatically sound but it's very pretentious.)

So, the choice is: Do you try to sound natural or do you speak correctly?

I'm afraid it's not for me to answer that. Personally, I would say "Who am I speaking to?"

I know it should be whom and I know I'm ending a sentence in a preposition, but getting those wrong is better than the "fix" if you ask me. The fix sounds awful.

How about:

Can I ask who's speaking?

Good question. It's not often I recommend deliberately getting the grammar wrong!

Here's a lesson on who and whom.