KATE said on twitter he wrote the music for this with James Harris:
https://twitter.com/hagakeita/status/796556998959661056
Quote:
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『Fate/EXTELLA』本日発売です！今回もハリスさんと共に音楽をやらせていただきました。EXTRAシリーズ らしい曲はもちろん、FGOライクなもの、新しくチャレンジしたものなど色々入ってます。是非遊んでみてく ださい！
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No enclosure so maybe they're planning a proper release?
On another note, the new music he writes from time to time for Grand Order is very nice. I really hope they get around releasing that too.