Doubling the frequency utilises the core twice as often, so it can be half the mass, but the heating due to magnetic losses doubles. A smaller transformer runs hotter because the heat exchange surface area is reduced.
The wire is shorter because it surrounds less core, so the windings weigh less, have lower resistance, and so can have less cross-section area. At a higher frequency, skin effect indicates thinner but wider copper strap windings, or a finer litz wire.
The core material may need to change to reduce magnetic losses at higher frequencies, but higher frequency magnetic materials have a lower μr and lower saturation threshold. All is not lost because the inductance of the transformer must be reduced proportional to the rise in frequency.
The problem with estimating the change in size is knowing and juggling the balance between magnetic losses and copper losses. The operating temperature of the windings should be similar to the core. If that is not the case the design could be optimised further.
From 60 Hz to 1 kHz the factor is 16, which is not small, so more change will be required and prediction ratios will not hold. Skin depth will change by a factor of 1/√f = 0.25 There is no advantage having unreachable copper conductor, nor deep magnetic core material.
There is no question that a new 100 kW design would be a challenge, but it has been done several times in the last 5 years. Those products are out there.
There is a possibility of running 2 or more, smaller, faster converters in parallel. That would give some modular redundancy, distribute the heat better, and make the converter less of a lump.