How are the billable costs of the architect determined?

As already mentioned, since you have only commissioned the architect for service phases 1-4, his cost estimate applies and not less. Your "offer phase" does not exist for him.

You did award him a contract based on an offer. The fee should be regulated there! That is exactly what you will have to pay.

Now my question: is the architect allowed to include the costs for the shell construction according to the HOAI in his billable costs if we assign it ourselves? I

If you were well informed, you wouldn’t even think of asking such a question.