Driver responsibility assessment fee is imposed, in addition to the fines and surcharrges, whenever you have six or more points within 18 months, or driving at least 21 mph over speeding limits, according to NY traffic law.
NY DMV cannot suspend your CA license. It can, however, suspend your driving privilige using in the state of NY. What is going to happen is that if you do not pay this fee, your driving privilege will be suspended in NY. The NY DMV will send a notice to the CA DMV about the suspension, under the driver license compact, which both CA and NY are signatories. CA DMV, then, will assess the violation under CA traffic law, and impose either points under CA law, which your insurance will increase, or suspend your CA license if you have enough points.
If your CA license is suspended here, then a regular APS hearing will not result in an reinstatement because the suspension was incurred originally from NY. In another words, if your license is suspended by CA DMV, then you must clear the NY matter, because you can have an APS hearing to reinstate your CA license, which is a mess.
So the bottom line, there is no way to contest now since you have already pleaded guilty on the traffic ticket and paid the fine. The assessment fee is valid under NY law. You will have about 30 days to pay them or it will be a mess.