I'm no expert on the topic but I believe that the Pi will still place a load on your ups after it is shut down. Furthermore although you can't boot up your Pi using your pico ups what you could do is place a button between the ups and the Pi to allow you to temporarily cut power to the Pi, which would boot the Pi. Anther possible solution to your problem and this is the solution that I would recommend is that you google tutorials on reducing your pi's power usage so that you can have it run for longer on battery instead of having to turn it off.
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