What is the security value of this on-the-fly user registration?

I am currently implementing SRP on a game I am developing. However, I will be distributing this game server and client, so I would like registration to be allowable on-the-fly, without requiring the host to use some sort of forum or email communication to pre-register the account before the first login. I'll go over the basic flow of SRP, and then my registration changes.

SRP Documentation: Found Here

1. The User enters their username and password into the client.


2. The client sends the username and a randomly generated value to the server.


3. The server gets the salt and password verifier (which is the result of a function on the password, both stored externally) of the user and sends the salt and another randomly generated value to the client.


4. Each side now independently calculates a strong shared session key, and proves that key to each other.


5. If both proofs match, the user has successfully authenticated.

To add the registration functionality, here are my proposed changes.

1. The User enters their username and password into the client.


2. The client sends the username and a randomly generated value to the server.


3. The server checks to see if there currently exists a user with the username sent from the client.


4. If no user exists, the server sends the RSA Public key to the client, requesting the client to generate a salt, and verifier.


5. The client sends the salt and verifier to the server, and encodes it with the public key.


6. The server decodes the salt and verifier with the private key.


7. Each side now independently calculates a strong shared session key, and proves that key to each other.


8. If both proofs match, the user has successfully authenticated.


9. If the user has authenticated sucessfully, the salt and verifier are saved to the database, or file for the user.

Would this be an acceptable, secure form of on-the-fly registration? Or are there some things that I haven't considered?