As a TPU at drill I severely injured my hand. What are my unit's options as far as putting me on orders until my treatment is complete?

That, I think would be up to your unit...if they are even able to. If you injured your hand during BTA, there certainly needs to be documentation, like an LOD, so that you can get treatment on your hand via the Army's dime, and also into your military medical records for when you get out and want to claim VA disability.

Update: I found this link. It may or may not help/answer, but it's worth a shot

https://wct.army.mil/modules/soldier/s6-extension.html

You were injured at drill, you generally don't go on orders to get treatment, unless the injury is so severe that you need to be inpatient in a hospital for more than 30 days, then that hospital is a WTU, like at JBLM.
This isn't an option for every kind of injury, there are some specific medical criteria to meet.
You can apply, and if you get denied, then you move on to the following:
You get an LOD approved to bill Tricare for followup treatment as authorized.
If the injury also prevents you from performing your civilian job, you apply for INCAP, a calculated percentage of your missed wages at your civilian job.
Those aren't options for your unit, those are both regulatory requirements.
And you owe your unit some paperwork for each of those.
If your UA cannot figure those out without help, he can call your BN MED NCO or S1.
And your UA should have already initiated the LOD during the drill, when you were injured.

Make sure that the unit does an Live of Duty and commanders inquiry and you receive documentation regarding the injury and recovery. Don’t expect to be placed on pay status though.