I've had bullfrogs since I dug my first pond last year in May. I've not had any fish go missing, although I'm sure there have been some small ones that I never saw in the first place. I know that bullfrogs CAN eat fish that are almost as big as them, but they typically will eat bugs, grasshoppers, etc., or things that fly by or jump out of the water. I enjoy feeding my bullies grasshoppers, the bigger the bullfrog, the bigger the grasshopper I toss in. I make it fair, if the grasshopper is fast and smart, they get right back out of there. I always toss it onto lilies. Some are really smart, no way do the bullies get to them. I'm not totally cruel.
So, in my opinion, not only will they eat your bugs, but if they spawn in your pond, they will provide food for your fish, too! Not sure how big your fish are, but I have goldfish (all sizes, from less than 1" to about 6") and koi (babies are about 4" now, largest is about 14") and of the fish I can identify, have lost none.
Good luck with your frog! As Addy said, even if you rehome it across the street, it or a relative will be back. Going to be hard to keep them out of your pond. So, might as well let them help keep your yard bug-free!