Your question has a lengthy introduction, but finally:
If at some point I will need to prove the authorship of a certain publication that bears a name slightly, or significantly different from what is in my documents, will that be a problem?
No. A slight difference that can be explained by differences in transliteration is not a problem. This is not a court case, people working in academia are human. Some people even change their name completely (in some cultures this was or is common upon marriage) and still they can list articles under both names on their CV.
I'm not sure exactly what you mean by significantly different because that doesn't apply here. Just use the transliteration you prefer.