Qiskit / qiskit-ibm-runtime / 28425034149

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Use backend's api client during QiskitRuntimeService._run (#2931)

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### Summary

Revise `QiskitRuntimeService._run` so, instead of using the "active" API
client (which can be modified by other calls - see #2771), it uses the
API client that corresponds to the backend's instance.

### Details and comments

Fixes #2892

### AI/LLM disclosure

- [x] I didn't use LLM tooling, or only used it privately.

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Co-authored-by: Samuele Ferracin <sam.ferracin@ibm.com>

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