You will be glad to know I solved the puzzle! Turns out I was fundamentally misunderstanding a game mechanic, and that I've been breezing through previous puzzle sequences without help and without the game's helpful items, which has then left me assuming I've been using them when I haven't. Also, it makes me quite proud that I was a few seconds off beating the puzzle without any help items (which is impossible, as I know from always ending up out of time in the same spot). Had absolutely buckets of time left as soon as I worked out what was going on!