The PM should escalate the issue of the perceived requirements change to the change control board and governing body, including representatives of both the client's and supplier's contracting folks, to iron out and come to an agreement about those changes. The client is only one party to this two party agreement so they cannot simply dismiss this accusation as different interpretations. They have an obligation to sit back down at the table and argue their case, and hear the supplier's case, hopefully to arrive at a mutually beneficial result. Even if only 50% of the perceived change results in a retroactive change order, then the team will get some relief in time and cost.
The project sponsor can have all the expectations he wants. The hard truth is, if targets were placed competitively on a project's time distribution, this project had a good 40% to 50% likelihood of overrunning its time target, and the sponsor needs to deal with that as an adult. Things happen outside of a capable PM's control, including stochastic events--pure chance--that both favorably and unfavorably affect a project's outcome.
No matter the outcome of the change effort, the PM needs to report factually the project's current health and predicted outcome, alternatives, and recommendations, and then let the sponsor make some decisions, let the chips fall where they may.
(I do not know the contract type under which this project was let, so this plays into it, too. If it was a firm fixed price deal, then time is the only piece to discuss. However, it sounds like this is not the case since the sponsor apparently had a vote in additional resources. If it was FFP, I would still pursue the retroactive change piece but I'd also bring in additional resources to try to crash the schedule if I thought that had some likelihood of working.)