Python base64 hash with same output as C# Script

I'm working with a vendor who owns an API. In order to call the API, they require us to hash the whole body of the request and add it to a Content-Digest digest header key. Content-Digest: SHA256=<digest>. They have provided us with a RSA Private Key along with a linq LINQPad file written in C#. This script is what outputs the base64 encoded hash that goes into the Content-Digest.

The problem is, I don't know any C# and the application that we are going to be using this API for is written in Python. What I'm looking for is a way to output the exact same formatted hash in a Python Script.

This is the C# Code they provided:

void Main()
{
    var payload = GetPayload();
    SignData(payload);
}
private void SignData(string payload)
{

    var keyFormatted = GetRSAKey();

    byte[] privateKeyInDER = Convert.FromBase64String(keyFormatted);

    var rsa = DecodeRSAPrivateKey(privateKeyInDER);

    var data = Encoding.Default.GetBytes(payload);
    using var hasher = new SHA512Managed();
    var signBytes = rsa.SignData(data, hasher);
    var computedSignature = Convert.ToBase64String(signBytes);
    computedSignature.Dump();
}

private static int GetIntegerSize(BinaryReader binary)
{
    var bt = binary.ReadByte();

    if (bt != 0x02)
    {
        return 0;
    }

    bt = binary.ReadByte();

    int count;
    if (bt == 0x81)
    {
        count = binary.ReadByte();
    }
    else if (bt == 0x82)
    {
        var highbyte = binary.ReadByte();
        var lowbyte = binary.ReadByte();
        byte[] modint = { lowbyte, highbyte, 0x00, 0x00 };
        count = BitConverter.ToInt32(modint, 0);
    }
    else
    {
        count = bt;
    }

    while (binary.ReadByte() == 0x00)
    {
        count--;
    }

    binary.BaseStream.Seek(-1, SeekOrigin.Current);

    return count;
}
public static RSACryptoServiceProvider DecodeRSAPrivateKey(byte[] privkey)
{
    byte[] MODULUS, E, D, P, Q, DP, DQ, IQ;

    // ---------  Set up stream to decode the asn.1 encoded RSA private key  ------
    MemoryStream mem = new MemoryStream(privkey);
    BinaryReader binr = new BinaryReader(mem);    //wrap Memory Stream with BinaryReader for easy reading
    try
    {
        var twobytes = binr.ReadUInt16();
        if (twobytes == 0x8130) //data read as little endian order (actual data order for Sequence is 30 81)
        {
            binr.ReadByte();//advance 1 byte
        }
        else if (twobytes == 0x8230)
        {
            binr.ReadInt16();
        }       //advance 2 bytes
        else
        {
            return null;
        }

        twobytes = binr.ReadUInt16();
        if (twobytes != 0x0102) //version number
        {
            return null;
        }

        var bt = binr.ReadByte();
        if (bt != 0x00)
        {
            return null;
        }

        var elems = GetIntegerSize(binr);
        MODULUS = binr.ReadBytes(elems);

        elems = GetIntegerSize(binr);
        E = binr.ReadBytes(elems);

        elems = GetIntegerSize(binr);
        D = binr.ReadBytes(elems);

        elems = GetIntegerSize(binr);
        P = binr.ReadBytes(elems);

        elems = GetIntegerSize(binr);
        Q = binr.ReadBytes(elems);

        elems = GetIntegerSize(binr);
        DP = binr.ReadBytes(elems);

        elems = GetIntegerSize(binr);
        DQ = binr.ReadBytes(elems);

        elems = GetIntegerSize(binr);
        IQ = binr.ReadBytes(elems);

        // ------- create RSACryptoServiceProvider instance and initialize with private key -----
        RSACryptoServiceProvider RSA = new RSACryptoServiceProvider();
        RSAParameters RSAparams = new RSAParameters
        {
            Modulus = MODULUS,
            Exponent = E,
            D = D,
            P = P,
            Q = Q,
            DP = DP,
            DQ = DQ,
            InverseQ = IQ
        };
        RSA.ImportParameters(RSAparams);
        return RSA;
    }
    catch (Exception ex)
    {
        Console.WriteLine(ex.Message);
        return null;
    }
    finally
    {
        binr.Close();
    }
}
private string GetRSAKey() {
    return "private key";
}
private string GetPayload()
{
return @"{
""key"":""value"",
""key2"":{
    ""subkey"": true,}
    }";
}

And it outputs something like this (344 Characters): 15vgzyv8Cke3Mkkwc3ryAgDMmY6olRfvgLqNyfhfti2GAfLb6s/vgrWc1p5jlWgQHh37Ir7UYThXldspriBz5NPl+BSFIW2dxXTMO2NMpzgc/5fmFN2maJCgwzDP0aqupmUGrw/DZp8zMAKtxWqs+8TGQTDthAW+4Y8g0hoLYSTEIHwvbkBUCspWo4Qr0MXj86P1Gsu5DbQ4Fs23fbajPuZqRHTyYzeANvxnma9mm30CwLD6blnKOLa+xRVd6eeuHu+Hp+F8hl5xSJS0Bcse4K0ZKccDD6sm4KSX2vaNQeQQ45fIDYLRUXYckGifqu7nJLwHILEenxue10841IHleA==

I've been trying to figure this out in Python and this is as close as I've got:

import hashlib
import base64
import hmac

key = '''
private key
'''
msg = '''
{"body to hash"}
'''
print((base64.b64encode(hmac.new(bytearray(key.upper(), "ASCII") , bytearray(msg,"ASCII") , hashlib.sha512).digest())).decode("ASCII"))

However, this returns a hash that looks like this: 9dfgSdEzLEdGHze/SrYCSGVHurEvFabe3YgBSqKowxHb96UznenFFoeTDjx2dlk2B53qq9ISKVwv+xFBXMBePQ==

If there is anyone who can help, it would be greatly appreciated!