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When you pull the plunger back you can see (clear tube) air bubbling out of the calliper, along said tube, into the syringe. It’s clear that it’s coming from the calliper (Fenella can confirm this).
What’s interesting is that on the push, air doesn’t go up the brake line and emerge from the lever. Which makes me think that maybe air is being sucked in past seals in the calliper. I’m increasingly suspecting the banjo, possibly due to Mumford and Sons.
How good is the syringe you are using? Whilst they may seem to be able to compress air, surely they are not designed to be pneumatically tight? Could air be leaking past the plunger? And (in before "everyone uses syringes"), that coupled with either less than perfect technique (soz) and particularly stubborn brakes = bike mechanics in the hotel car park.