Is it obligatory to complete the fast, or is it permissible to break it for someone who intended to fast but did not wake up until the Fajr Adhan, and did not have Sahur?

The matter of intention is easy. Whoever conceives in his mind that he will fast tomorrow has made the intention. If you intend to fast to make up for missed days, your fast is valid and sufficient, as long as nothing occurs to invalidate it. Hesitation in the intention to break the fast does not invalidate it; rather, it is definitively deciding to break the intention that invalidates it. There is no blame on your husband for not observing a supererogatory fast, and he is permitted to break it after commencing it.