Can JMeter and LoadRunner runs the API test script developed in Kubernetes?

Both tools are backend-agnostic to wit from JMeter and/or Loadrunner perspective there is no difference where the system under test is deployed: on premise, in the cloud, on the bare metal, in the virtual machine, in the container (the latter one is exactly your case)

You don't need to have in-depth knowledge of Kubernetes deployment architecture for creating and running the test, you might need to have an insight when it comes to results analysis in order to be able to answer the question: where is the bottleneck and what needs to be done to resolve it.

Kubernetes deployments are supposed to be auto-scaling so it will be a good idea to conduct a scalability test however it depends on your application deployment specifics. Just make sure that the containers performance monitoring toolchain is in place, the most popular solution is Prometheus however the other options could be possible starting from manually collecting OS/containers performance metrics, using JMeter PerfMon Plugin or going for an end-to-end APM tool

Both JMeter and Loadrunner support API testing so I fail to see why would you need both tools, it would make more sense to stick to one you have more experience with or where it's faster/easier to implement your test scenario(s)