I am planning an encounter for 4 level 2 players and I am planning to have them level up mid combat. I wanted to describe it sort of like a power within awakening like many popular animes have. To do this, I want to balance the encounter for 8 level 2 players, and when they level up they will have resources back to full like after a long rest. I want them to level up mid fight after spending some resources and realize the situation is dire.

What kind of balancing issues will arise from this? Are there any game-breaking facts I am missing that will ruin the game?

I'm sorry if this question was answered before, but I can't find it. I found this similar one, but it is from dungeon world and I don't know if there are similarities.
Can I make the Level Up move not require so much time without breaking the game?

Also I can't find rules about when to level up, or if the players have to rest to do so. If there are these kind of rules I would like to know about them too.

This should make the fight easier

You are correct that the xp allocation is similar between four 3rd level characters and eight second level characters. Be careful of things that could kill a 2nd level character outright (including using too many monsters). When the levelup occurs the party will be full-power 3rd level characters and the enemies will have, in theory, taken a few scrapes.

But none of this is the real problem.

This will draw out the game and interfere with pacing

Even for the most veteran party, levelup takes tens of minutes. Combat is already the lengthiest individual part of the game and your suggestion is going to add at least fifteen minutes, likely more.

I've forgotten to level up (I assume we all have) and only realized it when I needed a roll and it's very distracting; I don't get to enjoy what's going on the same way and I'm certainly not an engaged player. It's like talking to someone that's playing on their phone. (in case you were thinking of keeping the clock running, so to speak)

I highly discourage this. I understand what you're envisioning, but that's not how it's going to play out. There's no quick WoW-style "ding" to be had here.

If you're set on surprising your players with some sort of dramatic rallying effect, I would personally start throwing advantage out like it's candy (and use the XP guide for four level 2 characters). I haven't done this, so I can't speak to it, but the PHB tells us:

The DM can also decide that circumstances influence a roll in one direction or the other and grant advantage or impose disadvantage as a result.

and "a power within awakening" seems a lot like a circumstance that would grant advantage.

Alternatively, as Mołot pointed out in a comment, if you don't care about it being as much of a surprise, you can have them bring levelup'd sheets with them. Either tell them what it's for or be a little more vague about it "You guys might level up before the end of the night, and it'll speed things up to have it done ahead of time." You run the risk of people blowing it off forgetting to do it, but it's about the same as people not being prepared for the game in the first place.

Balance the encounter around 4 level 3 characters. If you balance it around 8 level 2 characters, it'll still be a very uphill battle and they will probably loose.

The big thing to realize about balancing a combat is that the number of actions and AC play a big roll.

1. If you make an encounter for level 3 characters, the combat will
   start out looking bleak, but when they level up and get absolutely
   every thing back the tail end of the encounter will be very easy.
   This should amplify the feeling of awakened power that you are
   seeking to convey.



2. 
   

   If you balance around 8 level 2 characters, the only real way to do
   that is with many small enemies with will throw the number of action
   the enemies have VS the party way out of wack.

   
   
   

   The other thing you could do is waves of enemies. That will allow you to adjust the encounter on the fly and let enemies keep coming at the party. The final wave only shows up after they have leveled up.

   
   

There is nothing "game breaking" about this, but the encounter itself could end up being either way too easy, or way too hard. It won't break any combat that takes place after this one. I would recommend not doing this for every level up because it will become a bit cliché. Level 3 is probably the best place to do it because that's when most classes dive into their archetype.

For your last question, the players do not need to rest to take advantage of the level up, they just typically do not regain full health and all of a sudden have prepared spells. If you are only planning to do this once, I think its fine.