I completely forgot your name.
The original Japanese sentence is more likely to be interpreted as “I completely forgot the name”. We usually insert お in front of a noun to show respect. Especially the sentence is ended with しまいました in a very polite way. In this case, dropping お is socially awkward when you talk about someone’s name, not the name of an object.
You can replace お名前 with あなたの名前, but the altered sentence sounds a bit indifferent and cold. お名前 has the nuance of “I’m so sorry but…”