That's what a bucket is for. You just use it in front of you for a second to breath.
It's worth the time to place those dirt blocks as manual demolition will take considerably more time. The ability to drop those walls in seconds is great. I learned this trick when building an underwater base before those monuments existed. Smaller scale than removing the water for a monument, so I would imagine it'd save even more time. If you do another underwater project I suggest giving it a try.