Clever approach, but there's a latency tax that hits hard at scale. Ran a similar setup - held up fine until we hit several thousand rows a day, then the import queue into Clay became the bottleneck, not the enrichment itself.
Other thing worth flagging: using Claude Code locally on unstructured company context works well for reasoning-type enrichment, but in my testing it's noticeably less reliable than Clay's native provider waterfall for verified emails and direct dials.
My recommendation - keep this pattern for anything requiring actual reasoning or summarisation, and stick with native providers for contact data. One thing that helped us was staging enrichment by data type: expensive verified stuff runs through the waterfall, contextual stuff runs through the sheet.