signtool fails to sign a binary with a key from a AWS CloudHSM

We are going to use AWS CloudHSM service to keep all code signing certificates secure and perform code signing on our build server. Our build server is Windows Server 2010, so I installed AWS CloudHSM client there. I activated cluster and all commandline utils work as expected: I am able to login, add keys, find keys etc. We would like to continue to use signtool to sign our binaries, so I thought that we can use key storage providers (KSPs) for AWS CloudHSM, that are installed along with other tools.

The Cavium KSP and CNG providers were installed successfully and are visible in the windows crypto provider's list. I defined environment variables as said here ( https://docs.aws.amazon.com/cloudhsm/latest/userguide/ksp-library-prereq.html ).

I added certificate via certutil to the HSM storage:

Certutil -CSP "Cavium Key Storage Provider" -user -importPFX  "certificate.pfx"
SDK Version: 2.03
Enter PFX password:
Certificate "myCertificate" added to store.

CertUtil: -importPFX command completed successfully.

Certificate has been added sucessfully and it appears when I execute findKey command from key_mgmt_util.exe console.

After that I tried to sign a binary with the certificate as it is said in https://learn.microsoft.com/en-us/windows-hardware/test/hlk/user/hlk-signing-with-an-hsm:

signtool_64 sign /n myCertificate "test.exe"

or

signtool sign /sha1 4F555EF9FAB8E86A2F84ACF325362A29FB64AF66 "test.exe"

but I got an error I cannot resolve

    SDK Version: 2.03
    Done Adding Additional Store
    SignTool Error: An error occurred while attempting to load the signing
    certificate from: C:\temp\test.exe

I also tried to specify key storage provider and key container

signtool sign /csp "Cavium Key Storage Provider" /k CARoot-877f51a1-90ee-4c10-8feb-02925caab4fb test.exe

that returned to me

SignTool Error: An unexpected internal error has occurred.
Error information: "Could not associate private key with certificate." (-2147024
891/0x80070005)

and

signtool sign /f certificate.pem /csp "Cavium Key Storage Provider"
/k CARoot-877f51a1-90ee-4c10-8feb-02925caab4fb test.exe

with other error message

SignTool Error: The specified private key does not match the public key of the selected certificate.

It seems to me that something is wrong with the certificate from the storage, but I have no idea how to fix this. test.exe exists on the disk and can be signed with signtool using certificate from another provider or when specifiyng pfx file.

What am I doing wrong? Is Amazon CloudHSM client compatible with signtool or how else can I sign binary on Windows using Amazon CloudHSM as a key storage?