Allow me to restate your question in the following sharper form:
Given two spinning tops with the same moment of inertia, but the second top with the center of mass higher above the supporting point, will the two tops behave differently?
I assume that at the start the spinning tops are as vertical as possible, and they start of course at the same spin rate. I also assume the two shapes are matched in such a way that the air resistance for both the tops comes out the same. I assume a normal friction for the point of contact of the spinning top.
After a while the top start tipping over visibly, and from then on things start snowballing.
As the top starts tipping a bit you get a torque from gravity. For the top with the higher center of mass this torque will be larger.
A larger torque has a larger corresponding precession rate. The kinetic energy for the precessing motion comes at the expense of the gravitational potential energy of the spinning top. That is, in order for the precessing motion to get going the center of mass must drop a bit. Pitching over provides that drop. All this means that for the same spin rate the top with the higher center of mass will pitch over a bit more than the top with the lower center of mass.
The point of contact of the spinning top is not an infinitely sharp point, it has a radius. As the top start pitching over the point of contact with the surface that it is resting on is no longer coaxial with the main symmetry axis of the spinning top. This gives rise to additional friction. The more the top pitches over the stronger the adverse effect of the additional friction.
So indeed you do expect that in normal circumstances, normal friction, the top with the higher center of mass will have shorter endurance.
As to why you cannot find a mathematical determination:
A possible explanation for that is that the usual approximation does not take into account that the center of mass must drop a bit in order for the precessing motion to get going. A common approximation is to set up the calculation as if the spinning top starts precessing instead of pitching.
The following two resources provide all the information that is needed to understand gyroscopic precession.
A 2012 answer here on physics.stackexchange, by me, to the question: What determines the direction of precession of a gyroscope?
Paper by Svilen Kostov and Daniel Hammer: 'It has to go down in order to go around'.
Kostov and Hammer discuss how a spinning gyroscope responds to a torque, and they have performed a benchtop experiment to verify the expected behavior