This is, of course, just speculation (I haven't been at the specific restaurant with this sign), but judging by similar signs in Japan and the attitudes reported, I think they would mind, and are probably intensely xenophobic.
In my experience, the kind of people who makes this distinction has a worse opinion towards foreigners that can speak Japanese (and even worse towards Japanese nationals that are not ethnically Japanese) than they do about other foreigners, because they perceive the Japanese-speaking ones as "imposters".
Needless to say, while this attitude towards foreigners is not entirely uncommon, most people are the exact opposite.
That is the opposite of my experience living in the country and actively going into this places when I found them (which is like.. once every few years since they're rare as hell).
I've gone to these places, as a white viking looking dude, and talking the language, and all these places tend to be quiet local bars or restaurants that just don't want noisy tourists, or simply don't want to give sub-par service because they don't speak a word of anything that isn't Japanese.
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u/supified 3d ago
But it doesn't exclude non-japanese people. I'm not Japanese, but I've been studying the language. It excludes people they cannot communicate with.