The tinge may be an optical illusion caused by refresh rate. Try connecting to the KVM via a VGA to VGA cable plus an VGA-->Mac adapter using various settings (the video card can only deliver 640 x 480 resolution but set the adapter to use higher resolutions, so that the refresh rate changes).
What happens when you connect directly from Mac to the monitor without going through the KVM?

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I get the same thing when connecting the IIcx directly to the monitor. I tried switching resolutions using a VGA>>Mac adaptor and it gave some interesting results. The green tinge disappeared when the adaptor was set to higher resolutions, but the Mac displayed at 512x384 and at an unbearably low refresh rate. The screen was flickering so much, I reckon it was refreshing at 40Hz or maybe lower.

I also tried installing the Basic Monitor Extension, but it didn't affect the display.

The monitor in question is an Acorn (unsure of the exact model), which is basically a re-badged Philips. It dates from approximately 1992. It seems to be capable of weird and wonderful things, like it can display at 640x480, 800x600 and also much lower resolutions which are used by BBC Micros, eg. 160x256, 320x256. When switching to these low resolutions, you can here the refresh frequency of the monitor change and it gives a slight high-pitched noise like a TV (some people can here this, some can not).

The Acorn A4000 uses 640x480 for most tasks but switches automatically to lower resolutions for certain applications and games.