"So, how can vBE be inversely proportional to VT?"
I think, this leads to a false understanding of the effect to be observed.
With other words: Base-emitter voltage does NOT decrease (automatically) with rising temperature.
The effect is as follows: For rising temperature the collector current Ic increases (because of Is temperature dependence). That means: To keep this current Ic on the same level the base-emitter voltage must be (externally !) decreased. Hence, the data sheet says that for constant Ic the well-known value −2mV/K applies.
I like to add that in the following link the temperature dependence of Is is derived as well as a formula for the „magic“ value of -2mV/K .
It is interesting to note that this derivation is based on transistor physics only - and without using the base current and the current gain at all.
In this context, I remember some - often controversal debated - questions whether the bipolar transistor is controlled by the current Ib or the voltage Vbe.
For me, the derivation contained in the said document is a further clear evidence that the transistor - physically speaking - is controlled by the applied voltage Vbe.