Signing the code doesn't help. Where are you reading the signature-verification code from?

Presumably you use TPM and the various trusted boot technologies.

I think you’ll still have the problem with this thing being on a pci-e bus though.

Yes, I was more curious if a malicious actor (the hard drive) with access to the nvme bus could manage to exfiltrate data directly via the network interfaces without involving the OS.

I suggest you look into TPM and the full signed-code architectures it enables.