The voltage of a battery droops during discharge because of its internal resistance, which is primarily a function of the ion concentrations in the electrolyte. Because ions must physically move through the electrolyte during discharge, a gradient in the ion concentrations is created, with them being most depleted in the vicinity of the plates. Pausing the discharge allows those gradients to smooth themselves out, reducing the overall resistance. This means that for a constant-current discharge, you can recover a little more of the energy stored in the battery to do useful work, because you are wasting less of it as self-heating in the battery.
However, if you have a constant-power load, as you note, you need to raise the discharge current between the pauses in order to maintain the same average power output to the load, which means that the power lost during the discharge periods could be higher than it would be with continuous discharge, even after accounting for the reduced average resistance.
Furthermore, you also need to consider the overall efficiency of the circuitry that allows you to pause the discharge. I don't know what you have in mind, but transistors have voltage drops, relays have coils that need to be driven, and the control circuitry consumes power as well. The extra complexity also has an adverse effect on the overall system reliability. My guess is that the losses will far exceed any potential gain from doing the paused discharge. You'd be better off just using extra batteries to reduce the continuous discharge current in each one.