If Egypt or Sudan was to seriously recognize a 3rd party claim to Bir Tawil, I imagine a condition would be that they recognize that the Hala'ib Triangle belong to the respective country, and you have to pick one.
That is my understanding as well. Neither side claims Bir Tawil only because doing so will weaken their claim on Hala'ib Triangle. I guess it will be Sudan that will object to this more since in de facto terms Egypt currently controls Hala'ib Triangle. So if Bir Tawil is given away then Sudan has no leverage.
