You are not entitled to believe that the terms “eternal” and “begotten” are mutually exclusive, because you are not the arbiter of what the words mean.
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The definition of eternal:
“without beginning or end; lasting forever; always existing”
The definition of begotten:
“Something is begotten when it’s been generated by procreation — in other words, it’s been fathered”
Christ referred to God many times as His Father. You can’t have “no beginning” if you have a Father. Christ says He is the Alpha and the Omega, or from the Alpha to the Omega, the beginning and the end. With God The Father there is no beginning, certainly at least in time, yet for Christ there is and He says so in His own words.
So, while I can believe anything I want that does not make it so, but the dictionary makes the terms mutually exclusive.
You did not consult a theological dictionary. You can’t resolve a theological question with a standard desk dictionary.
Consider where a dictionary or lexicon came from before you use it as the standard to define Biblical terms. There was no dictionary during the ages in which the Bible was being written. Words are defined by their use in context, not vice versa.