jremington:
You will notice that I mentioned TWO possible problems in reply #1.
I may have grasped it? It's the TP4056 causing the problem. The datasheet seems to say 1050mA max output, so I reckon its protection is shutting it down. My hunch is that by adding an electrolytic between the TP4056 and the boost converter, I'll solve it by providing a buffer.
I would have tested it already instead of asking on here, but I only have one boost module, and I foolishly got over confident a few days ago and it's hot glued to death so putting a cap between the modules may prove destructive. Before I embark on a potentially one-way endeavour... am I on the right lines here?