How do you handle power users who are burning through your resources on a flat-rate plan?

We launched with simple predictable flat-fee pricing model to make onboarding as frictionless as possible for early customers. It helped us secure our initial user base quickly, but now a small percentage of power users are consuming 90% of our cloud infrastructure and customer support time. Moving them to a usage-based tier feels necessary for our unit economics, but we are terrified of sparking a massive backlash from the very people who championed our product from day one.
What is the best playbook for shifting heavy users to consumption pricing without looking greedy?