Bones are fine, but..
Bones account for about 12% of the weight of the human body, and assuming the same would apply to a Kaiju (Which it doesn't have to), for the purposes of weight, it seems like they wouldn't have to generate that much negative weight.
A 100t Kaiju, with bones that are in direct -1g, would have bones that are minus 12 tonnes, meaning it would only weight 76 tonnes (Remember, the bones remove 12 tonnes, but their own weight is also removed, which is 12 tonnes on their own)
So that's already 1/4 less weight. Upping it to 20%, and you get 60t, less then 2/3 of the original.
So for the strict purpose of reducing weight, this would work, but bones aren't the only thing keeping a big organism alive
For instance, blood has weight too, and with the 8 fold increase in mass, comes an 8 fold increase in the volume of blood, which has to be handled by the heart.
I'm not sure about the mechanics of a heart suddenly handling all that, mostly because I think just as with the cross section you mentioned of the bones, the heart while 8 times the mass, can only handle 4 times the blood. Same with air.
..Maybe something else?
But maybe if the bones, or some other organ, generate negative mass in connected tissue?
Mass Effect fields, from Mass Effect, was essentially generated by Element Zero being given an electric current, and the configuration determined the strength, allowing negative mass, and thereby FTL travel.
But they didn't just use it for FTL travel. Engineering of buildings and spaceships also relied on it. A spaceship weighing half it's weight, would be twice as maneuverable with the same amount of maneuvering thrusters.
Effective control of such an organ, (like a muscle), could allow the Kaiju to increase it's mass when defending or attacking, or decrease, for increased mobility.
And most importantly, it would affect the weight of muscles, including heart and lungs, so that maybe, they have a fighting chance of moving all that air and blood around, now that they have to deal with less weight, and their own muscles can spend more energy on moving air or blood around when they don't have to support themselves quite as much.
As pointed out by Joe, lessened or negative gravity on blood or air would also work against the system.