How do I change battalions?

You don't. Your leadership decides when to move you, that's why it's called an assignment.
If you legitimately believe your leadership is toxic, utilize the open door policy of the next higher command. If your leaders are toxic, it's your duty to report it and help put an end to it.

If you want meaningful answers, you'll need to provide meaningful details.

-Is your assessment that the leadership is toxic, widely held throughout the unit? And, has it been held for a long time? Or is this the result of one incident, involving a very few individuals?

I already have some "red flags" that make me suspect.
-As a PV2 in the S-1 shop how and why are you in such close contact with your Company CO and your Company 1stSgt that you can determine their leadership is toxic? Where is the entire chain of command (Officer and SNCO) between you and them.
-Asking a question like this in a public forum, without making it a "hypothetical" indicates that you have a judgment problem most likely the result of being young, inexperienced, and possibly entitled. (I'm not saying this to be mean. You need to understand that because you don't like a leaders style, that doesn't make it toxic. It has been my experience that many young Marines that were having difficulty adjusting to military life suffered under the delusion that the chain of command was supposed to make sure they were all comfy cozy.)

This is the real world, not the sheltered world that mom and dad provided. Hitting the reset button and starting over is rarely possible. Unless, your toxic chain of command is doing something illegal, immoral, unethical or is so bad that it is harming readiness... you probably need to suck it up and resign yourself to making this better within the limits of your authority and your sphere of influence.

As stated by a couple of individuals, as a PV2 you are not likely to be transferred to another BN. You can make the request through your chain of command, but it is not likely to occur. There has to be a good reason for you to be transferred. Units don't like to give up Soldiers, and they are not obligated to transfer you, because you don't like the unit. If you believe that you are suffering from toxic leadership, then bring it up with your chain of command. Without being specific about what you believe to be toxic, you will not get very good advice. If there are legitimate concerns, and they are not being address by the chain of command. Speaking to the IG could be a next step. As SGT Miller mentions, you should watch your tact and cussing on the internet. Insulting other service members is not very professional, and potentially can cause issues for you down the road. It is not really hard for anyone on here to reach out to your chain of command at the 10th MTN DIV if they were so inclined.