Can my unit restrict me from taking a certain amount of leave before a PCS?

It's a PCS. Your commander is the approval authority for leave. You're not automatically authorized any amount. You're only authorized travel time.
Out in your leave form plus request PTDY. Assuming you have 30 days of leave, you can send off in April.

You can request leave. The commander can deny it.
If you are wanting leave prior to the PCS solely to avoid NTC, or to take a vacation prior to reclass, the needs of the Army (unit's NTC mission) takes priority.
However, it may be worthwhile to contact the gaining installation to check on the availability/average wait time for housing, and then the school so you'll know what the process and timing normally experienced (save the name and phone number of any contacts). This may provide information to support your request, of your intent is to arrive early enough to search for, and secure, housing.

Absolutely, the unit can restrict leave and when it can be taken, even in conjuction with PCS. The ONLY way this becomes problematic for the Command is if the school you are going to has a published training schedule which will allow no opportunity for leave prior to 1 October AND you have use/lose days which will be lost prior to 1 October if you do not take them prior to reporting to the school. And even then, it just becomes paperwork for the Commander, not a requirement.

Summary: Leave is an entitlement. You are entitled to take it, and the Commander cannot prevent you from EVER taking leave. However mission requirements ALWAYS trump leave, and the Commander can absolutely set restrictions on WHEN and HOW MUCH leave can be taken.

If you are PCSing away from the unit, an argument can definitely be made (probably) about how critical you, as an individual, are to the mission at NTC, but that is an entirely different conversation.