Confusing situation in thrust and conservation of momentum

The increase in velocity of the fluid isn't due to the frustrum pushing on the fluid. It's due to pressure differences in the fluid. Therefore, the fluid speed up will not accelerate the frustrum further upon release.

You can see this by considering the walls of the frustrum. Collisions of the fluid with the walls will push the fluid forwards, and thus those collisions push the frustrum backwards. Therefore, those collisions cannot be responsible for increasing the backwards velocity of the fluid.

This is a rough diagram of the same water element at subsequent times as it passes through the frustrum. As you can see a gap is created as water flows through the frustrum. Remember that in the frustrum's frame, all the water outside of it is flowing at velocity $v$. So the gap cannot be filled with fluid coming from behind because fluid behind the element outside of the frustrum is moving at the same speed as the element. On the other hand fluid coming from the smaller hole is travelling at a greater speed than $v$ while the velocity of the fluid outside the smaller hole just when the frustrum started moving is $v$. Because the fluid is incompressible therefore there is an excess of fluid coming out of the smaller hole that need to occupy some extra volume to preserve it's incompressibility. So what these excess fluid do is rush backwards over the outer surface of the frustrum to fill the gaps shown in the picture.

If in a time $dt$, $dm$ mass of water comes out of the smaller hole, then $dm$ mass of water will enter the gaps. But the velocity with which they enter the gap is higher than the velocity with which they leave the smaller hole. The idea is the same as the proof of $A_1 v_1 = A_2 v_2$ for incompressible fluids. The "area" provided by the gap is much much smaller. So to conserve momentum, the frustrum must actually gain momentum in the backward direction which will cause it to slow down.

Edit

Magnified diagram showing the gap. Gap is the region coloured in black. The extra region in blue inside the frustrum that is ahead of the outside water element previously occupied the black region when there was no frustrum.