My understanding was that the Correctional Service of Canada would be responsible only because it's responsible for the administration of the act. But you're correct that there's nothing in this bill that indicates who is responsible for making the payment.
The court that's making the order against the crown wouldn't necessarily be aware that these obligations were outstanding. It might not be feasible to expect the court to indicate that various portions of the award be paid out first to outstanding family lawyers and then to restitution and surcharge, and so on.
For the easier administration of the regime in an order against the crown, it's likely that whoever's responsible for making the payment—CSC in an offender's claim—would then be responsible for designating the portion or amount to be paid to the various debtors.