Nick,
I can’t give specific advice on this particular pump. However, pumps usually have a suction limitation as to how high they are able to lift the fluid before it arrives at the pump (this is limited because the pump is creating a partial vacuum in the pipe, thereby using atmospheric pressure to push the fluid up to the pump).
From what you’ve said, it sounds as though the pump works if it’s not having to lift the water out of the bottle.
You might be able to relocate the pump lower down, reducing the suction head required.
Also, try priming the pump (filling the supply pipe with water) - that may also help.
Drawback with this is that if you run the bottle dry, you’d manually need to re-prime the pump.
Hope that helps.