Fast TM on Mew?

Yeah, do raids and you'll get more fast TMs than you need, I had a dumb a few once before I had a Mew. I might as well have used them to play with a mew see what I got.

What are you hoping for? Be thankful your moves are both decent and are both super effective to 1 type. I would definitely not waste TMs on Mew if you don't have many.

Unless you have a bunch of tms and have a move in mind that you are shooting for I would keep it. It's a small chance of getting what you want without burning through lots of tms and there is a possibility of ending up worse.

After spending a bunch and trying many move permutations, I can tell you that there ultimately isn't much difference. Frost Breath/Wild Charge is a perfectly good set. I wouldn't touch it.

I'd look at two factors:

1) What Types do you personally need?
2) What Types are you normally fighting?

And a third question:

3) How many TM's are you willing to waste on this?

With only 1 or 2 you're willing to spend, I wouldn't bother changing anything. I'd call it your AAMew!Mew! (Anti-Air Mew! Mew!) and leave it.

It's a good set if you lack Raikou, though Zapdos can certainly do the job better (though those are in short supply) and Jolteon can fill the niche pretty well.

Frost Breath is one of the best Quick Moves for Mew - have you considered trying Charged TMs for double Ice? There are two Ice Charged Moves so you actually have double the chance of any individual move of getting it.

https://pokemongo.gamepress.gg/mewvesets-and-their-place-meta

Honestly depends on what you want to use it for. You could certainly do worse than FB/WC.

There was a small window where I brought my Mew to above 2000cp and got it Shadow Claw and Psyshock...however, then I took my best Latios to 30 and now working on Rayquaza number 2