I'm looking for bit/tiedown recommendations for a finished barrel horse. He is big and strong and fast. He has a very long neck and carries his head high and throws his nose up when he wants to evade the bit. If you tug on him he will just fight more. He wants to blow out of the first and second and run home. He did not do this with original owner. The people I bought him from admitted he scared them and he developed some bad habits. I'm curious about no hit bits and simplicity bits. 

Have you tried a combo bit? 

Also, I'm sure going back to basics and slow work wouldn't hurt either. 

Honestly, you can change bits but a horse can learn to push/brace against anything .... if nothing else changes.

This horse needs to learn to accept bit pressure. Period. That is going to come from your everyday riding and slow work at home. There's holes in his training (blowing out of barrels) that are the bigger issue - not the bit.

It's training. Not the bit. Just because he is "finished" doesn't give him a green card.

So, without more details in your post (how long you've have him, what slow work or retraining you've done, etc etc) or without providing a video, it's really going to depend on if you should maybe even stop running him all-together until some of these holes are better ....  or what to try.

What bit you chose depends on both him and you. I myself don't do well at all with big GAG bits. I do better in a bit that I could also show in (fixed shanks without gag) but that's just what works for me. If a horse is really, really pushy for slow work, I might even put them in a high port correction bit for a while, because I never want to be in a tug of war wtih my horse. I want them to respect what is in their mouth. If I have to go to something stronger to do that at first, so be it. (But I also will put my "soft"  horses in it too, because it's good for them to be able to carry whatever is in their mouth. Just good training.)

You as the rider also need to be extremely clear with your signals. Know when to hold. And know when you immediately need to release that pressure. If you are too soon or too late, the pulling and the bracing will continue. (That's where this is a training issue - not a bit issue.)

So I guess people can give you a long list of bits to put in his mouth but you likely already have something in your tack room that you can work with. Because these issues start at home, where you fix them first, before you go to the barrel pen.

Have you had him vetted? If not, maybe start there

Check out lesvogt.com he has some good information on different bits 

I'd start by having him vetted and have a dental done on him. 

then I'd try a combo bit and then a Crossway 

Listen to r_beau. Cut your losses and sell. No bit will fix it. God knows I tried everything because I'm hard headed. Wasted years of my life and almost quit racing.

Check your head gear, if bit is too tight it can cause the high head, if curb is too tight it can cause high head. And most of the time its pilot error/issues..Hands too high or to strong..

An old Montana cowboy once told me that when a horse puts its nose up in the air his brains flow right out his ass. 

Back to basics to fill the hole. JMHO.

Try a Jim Edwards bit. I noticed the flattened medium port kapt my daughter's head level.  You might be able to call and talk to him.  He's a wealth of information.