There were three possible approaches to take in regards to creating a Jewish state in the Middle East:
1. A South Africa style state from the beginning where the country is run by and for Jews and Arabs and anyone else are explicitly second class citizens or lower, and in this case it woudln't matter if Jews are the minority. Such a country could include Trans Jordan.
2. An essentially Arab Middle Eastern state with religious liberty and Jews are given guaranteed rights to settle. The closest something like this has actually happened in the Middle East is Lebanon, which had a really complicated power balancing arrangement. Such a country could also include Trans Jordan.
3. The Ulster approach of gerrymandering borders where the Jewish settlers were a minority. This obviously could not include Trans Jordan, and IOTL this was pretty much the route taken.
So including Trans Jordan means changing more than the borders, you have to change your concept of "Jewish state in the Middle East" from the start. Also, in the 1930s and 1940s the world was simply not going to get option # 2, colonialism was still very much alive and people really just didn't think in those terms. So you are left with #1, which would be an interesting but butterfly rich and controversial alternative history scenario.