Using encryption for authenticating messages (instead of signing)
Since everyone's answers were in the comments and I want to mark this as answered I'm answering my own question based on what I learnt from your comments and further research:

The answer is you simply can not encrypt with a private key. Encrypting with private keys is only possible in theoretical scenarios and only with RSA keys. 

And if you try to encrypt with the public key, then you will have to share your private key. And no, they are not interchangeable, most importantly if someone has the private key they can derive both keys from it.

So the only possible way is to sign the message.

This answer could be useful for people like me who have read simplistic online articles about how cryptography works and were led to believe that public and private keys are interchangeable.