I've been consulting for over twenty years at this point, and I've had my share of toxic clients (including one who actually sued me and was surprised when the judge wasn't amused by his suggestion I work for free either). You didn't say where you're from, where they're from (besides America), or what vertical you're working in, or what do you mean by "the end-justifies-the-means tactics" so I can only give the most general of advice here:

My experience is there are few VP who can stand up a project with an outsourcing firm without some risk of their own (either in their review, or in an extreme case: their career).

If that abuse is coming from people who report to your stakeholder, or from other people in the firm, a quick chat with the stakeholder would probably smooth things over. You might either find that obvious, or believe that it doesn't apply if you're getting abuse from multiple places, but it's not their money, just their job, so if you're not making your abusers' job harder in some way, your stakeholder will be in the best position to do something about personnel issues.

However if that abuse is coming from your stakeholder, you need to realise it's because they are receiving pressure, and you would do well to understand it because you might be able to help him/her with what they really need. Your stakeholder's job is to operate a P&L including your project against some revenue, or produce powerpoint slides for someone who does that job. Do those reports suggest this project is costing more than anyone expected? Try to attack the real problem here: Give your stakeholder some justification to go after the money, or to respond to the delay. This is what they really need.