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I use the Japanese input method (from the language pack) to solve this. Wish I was joking, but when in hiragana input, you're still able to input words from a Romanji base, and entire English input in full width characters if you start you word with a capital. As the IME has to compensate for having multiple words from the same keystrokes in Japanese, with Roman characters, you get accented characters in the same menu.

I don't exactly see this as a great solution, but with my limited use, it surprises me that this is restricted to a CJK input

have you tried https://github.com/samhocevar/wincompose

I haven't tried that one, I'll add it to the list. So far I've been using AutoHotKey and a script I found somewhere, which mostly works, but is a bit awkward to setup.

This looks promising, thanks for the tip!

The "international English" keyboard adds a few shortcuts to help.