Does voltage and current have an inverse relationship in DC power transmission?
the current being proportional to the voltage as long as the resistance is constant


This is the false premise: If you have a load that consumes constant power irrespective of the voltage, then that load must vary its resistance (higher voltage + higher resistance = same power).

P = I x U

and

R = U / I

both hold at the same time. So there is no way that you can keep the power and the resistance constant when either the voltage or the current or both change.

I.o.w., given R = U / I we can say

I = U / R

P = I x U = (U / R) * U = U² / R, so the power changes when the voltage changes, unless the resistance changes too.