First, please always cite a source when you quote something in your question. I had to do a bit of digging to find the original source--The Washington Post. Yes I know Post articles are usually behind a paywall, but you can still give information about the source site, date, title, etc.
It is true that "more of" is often used in conjunction with "than", forming a correlative conjunction. Here "more of" standing alone may appear a little bit unusual. Let's revisit the meaning of "more of... than"
used to say that one way of describing a person or thing is better or more accurate than another (Merriam Webster)
Certain words have been omitted from the sentence. Without omission the sentence should read something like this:
She wanted proof in writing that when it came to financial opportunities and inheritance, Barron would be treated as more of an equal to Trump’s oldest [four] children than their inferior (than a lesser child who falls behind in line of succession).
So the author is saying Melania Trump is fighting so that the scenario that her child Barron is treated as an equal heir to Trump's estate to other children is more likely to become reality than the alternative.
Therefore here "more of" does not indicate Melania wants Barron to have more than what other children, but equal. The use of "more of" just conveys a sense of slight uncertainty.