The classic Poisseuille flow approach is a fine approximate solution for situations that satisfy its assumptions. The effect of gravity can be accounted for well by including it in the pressure-drop term. It should work fine for water being sucked through a soda-straw.
Surface tension forces won't be large for water or milkshakes sucked through an ordinary soda-straw (~5mm dia.) Surface tension (and the two non-dimensional numbers you mention) would become useful in problems of fluid flow through porous media where liquid is flowing through capillaries.
I think milkshakes behave differently than water because they are non-Newtonian. When I pull the straw out of the milkshake, a thick layer (2-3 mm) clings to the outside of the straw ...
I can set a cherry on the top of my milkshake and it does not sink into the glass. That's not because it's floating (cherries are more dense than milkshakes) its because the forces in the milkshake beneath the cherry do not exceed the critical shear-stress required to cause flow. Below this critical shear-stress, milkshakes behave as a solid.
Note that the dimension of critical shear-stress (material property of milkshakes) multiplied by the characteristic length of the zone of yielding flow beneath the cherry just happens to have the same dimension as surface-tension. But that doesn't mean you're justified assuming the Bond or Cappillary numbers have physical meaning in this case. Dimensions may agree but the physics are different.
The Poisseuille flow approach assume the fluid behaves as a Newtonian-fluid. That assumption is likely violated for a milkshake. So Poisseuille flow solutions might be a poor approximation for analyzing milkshakes.