How to get constant current across varying Rload in h bridge
The best way to see what happens is for you to plot the base current of Q5 and Q6 as the voltage on the emitters (Q5, Q6) varies.  

You will see that the base current changes as the emitter voltage changes.  The only place that base current can come from is the current source.

Any current that is consumed by the E-B junctions of the transistors is not available to be sent to the load.

You also have other potential issues: if the magnitude of V1 and V2 isn't high enough, Q5 and Q6 will never turn off.  But you don't mention the magnitude of the voltage sources, so I'm just warning you about the potential issue.