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What I would like to get is something in 375 that comes close to the 9,3. Is there another rimmed case about the right size that might get me a bit more push?...maybe the 9,3 case would work

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The 9.3 X 74 R has I think a thinner rim than 375 Win by about 10 thou and that may cause some problems.
348 win is a much fatter case at the base than 375 win and would take out a lot of chamber metal.
348 actually has a larger base dia than 375 flanged mag.
If you are prepared to do a wildcat cartridge I would think the 405 Win would be a good cartridge to start necking down.
It has a slightly thicker rim and similar capacity to 9.3/74.
Another option to 405 could be 45/70 as the parent case.

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