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My LBS tried to bleed it, couldn't, and gave it back to me. They came to the same conclusion that I initially did, that something was broken inside the shifter so this forum has been much more helpful!
I actually bled it yesterday and rode it today, had a few issues with the bleed (pushing the lever all the way in, when opening and closing the caliper bleed nipple for example, seems to block the shifter up again) but I managed it and seems all good now :)
To be fair, I fixed the shifters to the bars myself when I swapped for a longer integrated stem/bar. There's plenty of hose between the frame and where it'staped to the bars but obviously not quite enough between the taping and the shifter and this pulling on the hose is causing the block inside the shifter somehow.