You're probably fine. SSH key-based login is designed to work over untrusted networks, without exposing key data - a VPN, even one which you don't control, doesn't change this.
At no point in the sign-in process is the whole private key sent across the network - this would be a massive flaw in the system if it was the case, and would make SSH key login useless over the internet.
In order to steal your key, the VPN provider would need to:
- Perform a successful MitM attack on your connection to the server (which would require access to your private key to be successful)
- Either inject a command to your system to read your private key, or coerce you into sending it to a remote point once they have a successful MitM in place.
If you were using third party VPN software, there would be a risk of that reading the key directly from your machine, but no more than the risk of any other software you run doing so.