Alternatively, from GCSFuse version 2.9.1 onwards, writes can be configured with streaming writes feature ( which doesnt involve staging the file locally ) with the help of --enable-streaming-writes flag.
But unfortunately, there are no volume mounting options in the Cloud Run console interface. And adding this option to YAML results in an error: Unsupported or unrecognized flag for Cloud Storage volume: enable-streaming-writes (field: spec.execution_template.spec.task_spec.volumes[0]).