A single Senator can prevent the suggested amendments from being even debated. It requires unanimous consent to move forward with those amendments. To do otherwise would violate Senate rules. Someone like Sessions or Demint can raise an objection to the unanimous consent request and stop the amendment from being brought up for a vote. This is the same thing Sessions did before the last cloture vote which failed. It can be done again.
Irregardless of Senate procedure, the $4.4 billion in funds is just a sop to get conservative Senators to buy off the amnesty bill as a whole. That will not happen as long as the amnesty and guest worker portions of the bill remain intact. Besides, are there not funds in the current fiscal year’s appropriations to build the border fence and hire more agents. If so where’s the fence and the additional Border Control officers. Has ICE been expanded in the last 6+ years. Have we controlled the border? Have we stopped the hiring of illegal aliens? No, No and No. Stop this nonsense. Build the fence, fine the illegal alien employers and employees, secure the border and enforce the law, period. Will that actually happen, I doubt it but it is worth a try.
Sessions has already publicly stated that he'll use whatever procedural tactics that are available to stop the bill and I believe him. He also said other senators are prepared to do so as well.