I want to accurately measure the net voltage drop on the load excluding caused by mosfet's RDS.
Short answer: Use a differential (or instrumentation) amplifier but, to improve its accuracy, you can filter the input voltage levels it receives.
I would also make one half of the full H-bridge define the polarity i.e. if you want current to flow from left to right (as indicated by the red arrow in your picture) then, the bottom right MOSFET in the bridge should be permanently activated with PWM generated by the left half bridge.
For reversed current the bottom left MOSFET should be permanently activated with PWM controlled by the right half bridge.
I say this because then it allows one (of the two required) voltage measurements to be fairly stable and, that stable voltage represents the volt-drop across the permanently "on" MOSFET.
At this point you could make an assumption that all four MOSFETs have the same voltage drop when on. Then, all you have to do is accurately measure the supply voltage and factor in the duty cycle of the switching side of the bridge and you can fairly accurately determine what the voltage across the SMA is.
To gain more accuracy, you could use a reasonably fast and accurate Instrumentation Amplifier (InAmp) with appropriate RC input filters (that remove the switching artefacts) and you should be good to go: -