What is the ruling on intentionally omitting the prostration of forgetfulness (sujood as-sahw), and what is the ruling on prayer when one doubts the number of rak'ahs, yet inclines towards one number and repeats the prayer thinking it is invalid due to omitting the prostration of forgetfulness?

Scholars differ regarding the ruling on the prostration of forgetfulness (sujud as-sahw). Some say it is a Sunnah, and therefore, prayer is not invalidated by its omission. This is the view of Ash-Shafi'i. Others say it is obligatory, but prayer is not invalidated by its omission, like the view of Abu Hanifa. Imam Ahmad went with its obligation if it was for something whose intentional omission invalidates the prayer. The apparent view of his madhhab is that prayer is invalidated by its omission if its place is before the salaam, but not invalidated by its omission if its place is after the salaam. There is also a narration from him stating that prayer is invalidated by its omission absolutely.

Ibn Rajab mentioned the detailed opinions of scholars on this matter. Among what he mentioned is that whoever said it is a Sunnah, prayer is not invalidated by its omission, such as Ash-Shafi'i and 'Abd al-Malik al-Maliki. As for Imam Ahmad, prayer is invalidated according to him by intentionally omitting the prostration of forgetfulness if its place is before the salaam, but not invalidated if its place is after the salaam, and it is not invalidated by forgetting to perform it in all cases. The other narration from him is: If one forgets it and a long time passes, one should repeat the prayer.

In your case, you were obligated to build upon the minimum, and only the Imam acts based on investigation and builds upon what he predominantly believes. Since you repeated the prayer in which you had omitted the prostration of forgetfulness, you have cleared your conscience and have exited the disagreement of scholars who hold the view that prayer is invalidated by its omission.