A better explanation is that the verb is محذوف here. So we should think of this phrase as actually being أبدأ بسم الله or أبتدئ بسم الله or أشرع بسم الله. All of these have been proposed by scholars as "understood" (محذوف).
I don't know which one is the best of them because I don't know if there is a hadeeth or pre Islamic poem in which the verb is actually mentioned explicitly. Until we find it, we cannot say which verb is the best to be understood (implied) here.
It's not corrupted -- the hamza is elided into a yaa' (a form of تسهيل الهمز), and this is permitted. It reportedly reflects the dialect of the Hijaz at the time, and is frequently seen in poetry. It should be بَدَيْنا, however, not بَدِينا.