I think the weight mass method you are proposing is good; but you will need to set a spring or something to make the thing retract when the weight goes up and calibrating that spring will be a difficult task. The floaters move freely. Floaters have been used forever to meassure liquids specific gravities and that maybe the reason why. Anyway if the weight (which is acctually a floater too) can be attached to the weighting system without affecting the meassurement then maybe it is another solution to the same problem.A basic rule in measurements is that the meassuring system used should not affect the system being measured.
What range of specific gravity are you trying to measure? I am thinking about lowering a heavy object into the liquid (not touching the ground) and having a force gauge hold the string. Depending on specific gravity the force gauge reads different values.
A heavy object will not be affected by changes in the liquid voyancy which is what's being measured