Answering my own question:

Thanks to @bestbeforetoday's comment, I managed to rewrite the signing code and now it looks like this, for anyone having the same problem as I had:

function getPrivKey(pemFile) {
  const pemContent = fs.readFileSync(pemFile, 'utf8');
  const key = crypto.createPrivateKey(pemContent);
  const jwk = key.export({ format: 'jwk' });
  const d = jwk.d; 
  return Buffer.from(d, 'base64url'); 
}

function fabricSign(message, privateKeyPemFile) {
  const privateKeyBytes = getPrivKey(privateKeyPemFile);
  const msgHash = crypto.createHash('sha256').update(message).digest();
  const signature = p256.sign(msgHash, privateKeyBytes);
  const signaturep1 = signature.normalizeS();
  const signaturep2 = signaturep1.toDERRawBytes();
  return signaturep2;
}