C# BouncyCastle - RSA Encryption with Public/Private keys

There are some errors in OP's code. I made few changes. Here is what I got running.

public class TFRSAEncryption
{
    public string RsaEncryptWithPublic(string clearText, string publicKey)
    {
        var bytesToEncrypt = Encoding.UTF8.GetBytes(clearText);

        var encryptEngine = new Pkcs1Encoding(new RsaEngine());

        using (var txtreader = new StringReader(publicKey))
        {
            var keyParameter = (AsymmetricKeyParameter)new PemReader(txtreader).ReadObject();

            encryptEngine.Init(true, keyParameter);
        }

        var encrypted = Convert.ToBase64String(encryptEngine.ProcessBlock(bytesToEncrypt, 0, bytesToEncrypt.Length));
        return encrypted;

    }

    public string RsaEncryptWithPrivate(string clearText, string privateKey)
    {
        var bytesToEncrypt = Encoding.UTF8.GetBytes(clearText);

        var encryptEngine = new Pkcs1Encoding(new RsaEngine());

        using (var txtreader = new StringReader(privateKey))
        {
            var keyPair = (AsymmetricCipherKeyPair)new PemReader(txtreader).ReadObject();

            encryptEngine.Init(true, keyPair.Private);
        }

        var encrypted = Convert.ToBase64String(encryptEngine.ProcessBlock(bytesToEncrypt, 0, bytesToEncrypt.Length));
        return encrypted;
    }


    // Decryption:

    public string RsaDecryptWithPrivate(string base64Input, string privateKey)
    {
        var bytesToDecrypt = Convert.FromBase64String(base64Input);

        AsymmetricCipherKeyPair keyPair;
        var decryptEngine = new Pkcs1Encoding(new RsaEngine());

        using (var txtreader = new StringReader(privateKey))
        {
            keyPair = (AsymmetricCipherKeyPair)new PemReader(txtreader).ReadObject();

            decryptEngine.Init(false, keyPair.Private);
        }

        var decrypted = Encoding.UTF8.GetString(decryptEngine.ProcessBlock(bytesToDecrypt, 0, bytesToDecrypt.Length));
        return decrypted;
    }

    public string RsaDecryptWithPublic(string base64Input, string publicKey)
    {
        var bytesToDecrypt = Convert.FromBase64String(base64Input);

        var decryptEngine = new Pkcs1Encoding(new RsaEngine());

        using (var txtreader = new StringReader(publicKey))
        {
            var keyParameter = (AsymmetricKeyParameter)new PemReader(txtreader).ReadObject();

            decryptEngine.Init(false, keyParameter);
        }

        var decrypted = Encoding.UTF8.GetString(decryptEngine.ProcessBlock(bytesToDecrypt, 0, bytesToDecrypt.Length));
        return decrypted;
    }
}

class Program
{

    static void Main(string[] args)
    {
        // Set up 
        var input = "Perceived determine departure explained no forfeited";
        var enc = new TFRSAEncryption();
        var publicKey = "-----BEGIN PUBLIC KEY----- // Base64 string omitted // -----END PUBLIC KEY-----";
        var privateKey = "-----BEGIN PRIVATE KEY----- // Base64 string omitted// -----END PRIVATE KEY-----";

        // Encrypt it
        var encryptedWithPublic = enc.RsaEncryptWithPublic(input, publicKey);

        var encryptedWithPrivate = enc.RsaEncryptWithPrivate(input, privateKey);

        // Decrypt
        var output1 = enc.RsaDecryptWithPrivate(encryptedWithPublic, privateKey);

        var output2 = enc.RsaDecryptWithPublic(encryptedWithPrivate, publicKey);

        Console.WriteLine(output1 == output2 && output2 == input);

        Console.Read();
    }
}

I was using an incorrect Public Key.. and the test that proved the Private and Public keys matched was using the correct Public Key.

The above code works perfectly as is, as long as you get the keys right!

I tried @Morio's solution but was getting some exceptions. The first one was

-----END PUBLIC KEY not found

Which I was able to fix by formatting both keys properly

publicKey = $"-----BEGIN PUBLIC KEY-----\n{publicKey}\n-----END PUBLIC KEY-----\n";

The second error was while trying to decrypt the text. Unable to cast object of type 'Org.BouncyCastle.Crypto.Parameters.RsaPrivateCrtKeyParameters' to type 'Org.BouncyCastle.Crypto.AsymmetricCipherKeyPair'.
Which I resolved by casting to the appropriate type.

var keyPair = (RsaPrivateCrtKeyParameters)new PemReader(txtreader).ReadObject();