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 understood that one is beginning with the name of Allah, it doesn't matter what verb is that, بدأ or شرع mean the same thing.
See this poem attributed to the companion عبد الله بن رواحة:
باسم الإله وبه بَدِينا
ولو عبدنا غيره شقينا
The part in bold is considered تأكيد for what was omitted.
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.