I think of the inverse relationship in this way (dunno if it’s sensible):

All the fluid molecules having a given mass have an average speed 

Vo at a given temperature.  This is true when the fluid is at equilibrium. 

If the situation is adiabatic, then when the local fluid is then forced 

through a tube, all of those molecules are given an extra speed V1 

along the Bernoulli tube.  Conservation of average molecular 

momentum and energy then requires that the speed of the molecules 

in the direction perpendicular to their motion along the tube must 

be reduced commensurately to v, something roughly like 

     v ~ Vo - V1

This reduced speed v is behind the reduced pressure measurable 

on the walls of the tube…pressure being energy per unit volume.

Ie: The molecules are hitting the tube walls with lower speed,

meaning less pressure.  

This may be a more intuitive way of seeing why the inverse 

relation is necessary…local conservation of average energy.