Can your unit make you volunteer as an augmente when they’re already deployed?

That's odd. If your unit had a deployment, and you did not extend to meet the requirements for that deployment, then you were supposed to receive a bar to reenlistment. That may sound harsh, but if not for that rule, about a third of the Guard and Reserves would just reenlist "around" deployments, leaving every DMD scrambling for names.
No one can "force" you to volunteer on MOBCOP, but if you don't, then don't be surprised when your BN initiates a 15-6 investigation into your extension after your unit mobilized, leaving you responsible for paying back any bonus or college money that came out of that...

Please go back and re-read your post. Before I get into it let me explain that I am a father too and I understand the pressures on you to be a dad. Being a father is the most important thing you will ever do. HAVING SAID THIS...You did reenlist, "So I can go to school and better my career for my family" As tough as this will be, and I will give it to you with the bark on it, the US Army, Active and Reserve components IS NOT there to help you better yourself, if you are not willing to do your part, and YOUR PART is to deploy when ordered to. Cut and Dried, if you want the Cake that comes with service, then you have to be willing to eat the other part too....Benefits are EARNED and it is by doing the Military job part of the military that earns you the benefits. In the military, especially the reserve components, if you are not willing to deploy when ordered, then what good are you to your unit? You take up a slot, train to do a job and may have troopers under you that depend upon you to be a leader, then when it comes time to actually do your job, OH Wait....I just joined for how this benefits ME...Sorry, I got better stuff to do than go on deployment....That is NOT how this works there Turbo.....
AS A FATHER, one of the most important thing that you will teach, impart to, or show by example to your sons and daughters is the IMPERATIVE of Integrity, Honesty, Honor, and Duty! What would your son think of you if when he is old enough to understand, you explain to him that you ditched your responsibility as a Soldier because it was inconvenient.
My wife was a drilling National Guard Soldier for her last three years of the eight that she served. Four months after our son was born, she came down on deployment orders, while I was on active duty. Her command gave her the opportunity to get out for family reasons. She, logically went through the thought process that she had taken the bonus, she had trained for the job and she had troops under her that depended upon her. She deployed for 18 months. I was a single dad for that time and we planned for me to go as well. Long story short, it would have been easier for our family if she got out, but the RIGHT thing to do was to fulfil and live up to the commitment that she made when she took the oath and enlisted in the Army. Remember as a Dad, the most important lessons that you teach your son will be by what you do not what you say.

Interesting. My understanding is going through mobcop doesn’t guarantee you’ll get the position you put your name in for.

I would think they would just assign you to a mobilization with their higher or sister units in the same command.

Going through mobcop you could end up just about anywhere.

The upside is it sounds like they would release you to go anywhere you wanted.