Caution Math! }:)
I think your math is solid, but as a practical matter; I already have my doubts that Dropships, especially are actually big enough as it is to do their job. The union is a good example; it's gotta be *tight* in their with 12 mechs and two fighters. Never mind the infantry carriers.
A Union is almost a football field in length, width, and height, but let's see what I have to work with. I'll also use the 3600 tons mass from
sarna, instead of the 3500 tons.
12 Mech bays and 2 fighter bays: 2100 tons
Engine: 19.5% * 3600 = 702 tons
Fuel: 216 tons
Weapons: 168 tons (not counting ammo)
Other stuff: ~475 tons (cargo, internal structure, fuel pumps, bridge, ammunition, escape pods, armor, crew quarters, etc)
So let's go with some volume estimates:
Assume each Mech is 12 meters tall, 6 meters wide, and 4 meters 'deep', plus a 3 meter padding around each mech to allow for Mech bay access, that is a volume of 18x12x10 or 2,160 cubic meters per Mech. Assuming the same per fighter, and you get a total of 30,240 cubic meters for 2100 tons of equipment, or about 7% the density of water (this is not counting the tonnage used for internal structure). To give you an idea of what I mean, put 2 Mechs in neighboring Mech bays, and move a gantry so that it connects the closest point of each Mech. You and a friend could lie down on that gantry, each with your feet touching one of the Mechs, raise your arms to each other, and you still wouldn't touch your fingers (assuming you are <=2 meters tall).
The Engine at 702 tons will have dense metal parts and empty chambers for the reaction to occur. I'll average this as 1/4 the density of water, so the engine takes up 2808 cubic meters.
1 kilogram of Liquid hydrogen is 14.128 liters in volume from
here. So 216 tons of it is just over 3 million liters in volume, or 3052 cubic meters. This will be slightly larger due to the fuel storage and pumps, but most metal pumps are more dense than liquid hydrogen.
Weapons have fewer empty parts, so they are 1/2 the density of water, or 336 cubic meters.
So 3186 tons have been allocated at a total volume of 36,436 cubic meters. The 80 meter diameter Union has ~800,000 cubic meters to play with, so it still has 95% of its volume available to fill with the remaining 475 tons of stuff.
The 80 foot wide Union is too small, as it only has ~21,700 cubic meters, meaning I have to free up ~15,000 cubic meters. The best way for me to shrink that would be to reduce the dimensions used for the Mechs. Reducing the volume around each Mech to 2 meters would drop the needed volume from 30,240 to 17,920 (saving me 12,320 cubic meters). Dropping that to 1.5 meters around each Mech reduces the volume needed for the Mech and ASF bays to 13,230 cubic meters, for a final total of 19426 cubic meters, leaving 2,274 cubic meters for the remaining 475 tons of stuff.
Another location I could drop volume is the engine. If that was changed to half the density of water, it would only take up 1404 cubic meters, and the freed up space would be used for walkways around it.
From the sarna page, the side-view cross section of the Union is 460 pixels tall (bottom of engine to top of dome), and at the Mech belt the effective room is 460 pixels wide (I am measuring to the narrower of the floor or ceiling distances, since the Mech likely stretches from the floor to the ceiling). The lower Mech bay measures 65 pixels to the top of the Mech Cubicles, and another 20 pixels to the ceiling.
So, I need a ratio of pixels to meters. Assuming 78 meters for height, that translates to ~5.9 pixels per meter. The mech bay is a cylinder 78 meters wide and assuming only the mech cubicles it is 11 meters tall, for a volume of 52,562 cubic meters.
However, there is that central elevator. I'll assume it is the square elevator seen in the color diagram, and get a volume used for that. The elevator is 120 pixels or 20.35 meters wide, and takes up all 11 meters of height. So the elevator takes up 4556 cubic meters, leaving ~48,000 cubic meters available for the Mechs, or ~4,000 cubic meters per. This is almost twice my original estimate of 3 meters of room per Mech needing 2160 cubic meters per Mech.
I'm also assuming that all the Mechs are stored in the Lower 'Mech Bay. If they are stored half and half, this doubles the volume for each Mech to 8000. If not, that means the upper bay could just be for the ASF bays.