Strategies for preventing abuse of API SaaS free-tier resources?

For business reasons, we don't yet have any paid plans (nor any of the infrastructure to support paid plans); so right now we're only giving our users access to a rate-limited free plan.

For the vast majority of our users, they can build and scale their use-case while staying well under the rate limits our free plan imposes.

But some of our users find our rate-limits constraining, and try to evade them by registering additional accounts, to then use multiple API keys in tandem to make their requests. (These users' apps are usually badly-architected, such that they're making 10-100x the number of requests they should be to achieve their use-case.)

Deactivating these users' "alt" accounts as we find them — or even banning these users altogether — doesn't help much, as these users just move on to using increasingly "stealthy" account registrations — using real-seeming email providers; registering through VPNs to disguise their traffic origin; splitting their requests by microservice such that their backend IPs/request-signatures never correlate; etc.

We could force users to go through a KYC/AML-like identity-verification process to register, but this would be a pretty massive hurdle to impose in the way of signing up for a SaaS product — and not one our users would expect, given that we're not a finance or gambling app.

Is there any other way?

Have you scaled an API SaaS with a free-tier plan and dealt with this problem? If so, what did you do?