Search for process instance via business key

Hello,
we are trying to find a solution for a following problem: We start a process instance with a variable which identifies a client. Later we would need to check if a process for given client is running. We are of course storing Camunda’s process instance id together with client identification on our side, but we need to cover the edge case when the update of our DB fails (but process is running).
In Camunda 7 it was possible to attach a business key to process instance which is not possible anymore in Camunda 8. We can create a variable with a client id and then use variables search endpoint. But that way we get all processes including finished ones. It is not possible to filter variable search by process state, so the only option is to get list of all processes via variable search and then make a call for each item in the result and check the process state.
Much better would be a search call directly for process instances where we could pass the client id and filter for state.

Is there any elegant solution for this ? To search for active process via variable value ?

Thank you

Best Regards
Robert Lufinka