That bubble is important. It means the pump is leaking, at least slightly, while off. With an ideal peristaltic pump, if you pumped until it has no air left in the line and then turn it off, it would indefinitely hold as if time had frozen. When you turn it on days/months later, it'd still be exactly as it was before, about to drip again. In reality they're not that perfect, and some leakage occurs. However, if you're seeing it have a bubble despite running it 4 times a day, it's leaking a lot.
What type of pump is that?
Average case that means one side drips out slowly into the tank and the other dosing container side drains back into the container. That tends to lead to erratic dosing, since sometimes it's pushing bubbles out, sometimes it's pushing liquid out.
If the dosing container is above the place it's dumping into, it seems plausible it's siphoning.
Regardless if it's the Kasa or the pump, I'd recommend trying to fix the pump. I'd pull the tube and see if it looks highly beaten up or worse case ripped/cracked; if so, replace it. Barring that, not sure there's much to do outside of getting a different design. There's a different thread where people are mentioning the jebao dosers are on sale for around $60. I'd personally trust those more than a kasa outlet and 2-roller pump that's draining that much.
One more question, if the pump was running at full speed, with all the lines fully primed, how long would you expect it to take to dump your full container?