I use O-ring bells (my model is BBB Loud & Clear Deluxe which has a very loud sounds, far better than many cheap bells). They can be mounted at any position of the handlebar. Usually O-ring bells are supplied with several different length O-rings and the mounting brackets for the O-rings have several positions, so practically any handlebar diameter between 22.2 mm and 31.8mm works, wrapped with bar tape or not. In the worst case, you may need to buy a differently sized O-ring but it shouldn't cost much.
You can even mount the bell on a stem if you don't have enough room elsewhere.
The position where I put my bell is generally over the tape ends. That's a location that's otherwise unused because you don't hold your hands at the position where bar tape ends.
If you want to ring the bell at the same time you brake, it shouldn't matter which side of the handlebar you mount it to, because it's not a good idea to brake hard and at the same time do something else (such as ringing the bell), so you can use either brake, front or rear, while ringing the bell because only really hard braking needs the front brake. One hand brakes, the other holds the handlebar near the bell. You may need to reduce the braking force momentarily when you move your hand to the bell, but when the hand is grasping the handlebar near the bell, you can increase the braking force again.