No, it is not possible. The fleet-server service account is only able to write to indices that are managed by fleet, and there is no way to change that.
You're definitely fighting against what Fleet/Agent tries to do.
It is designed to ingest into curated index names, and not be a universal data router in the way that Logstash is often used.
I don't understand why you want to have your logs routed like that, but it's not really a scenario fleet & agent are designed for.
The best suggestion I can make is to change your pipeline to something like:
"set": {
"field": "_index",
"value": "logs-{{{name}}}",
"media_type": "text/plain"
}
It's still likely to violate some of the assumptions fleet makes, but it's going to get you closer.