Dog water bottle that works with bike cage?

I am looking for a lid or something of the sort that fits on to a standard biking water bottle and can fit in the bottle cage on my bike. I have a Slurpy Sack 2.0 and love that the bowl flips up and down, that my dog can drink out of it easier and the fact that no water goes to waste. Is there a lid like this that fits onto a standard bike bottle?

The reason I’m asking is because I go on a lot of short rides with my dog and would love to store her bottle on my bike in the cage. The bottle we have now falls out of the cage because it is flexible and I haven’t found another bottle that works like this one.

Any suggestions?