A solar blanket is by far the most effective and lowest cost method to increase you water temperature and keep the heat in. I typically only cover my pool at night, and remove the cover during the day. I get a better heat gain on my pool that way, some say leave it on all the time so YMMV.
With that said, I seriously doubt that these clear covers will make a significant impact in heat gain, or retention. These solar rings can't cover 100% of the pool surface, leaving many gaps to allow evaporation at night (your main source of heat loss).
As far as ease of use, I can't imagine collecting 10 or 15 of these rings off the surface of the pool and storing them would be any easier than rolling up the solar cover. I have a cover and reel, and it takes all of 30 seconds to remove the cover, and about 2 minutes to put it back on.
I understand the issue wiht your pets, and you might be right about them getting in on top of the cover and possibly drowning.
Have you considred using an invisible fence around the perimeter of your pool to keep the dogs out? You could still remove their invisible fence solar to allow them to use the pool when you wanted them to.
One other possiblity for your is the liquid solar blanket (solar fish). They don't work nearly as well as a regular solar blanket (only about 30-40% as effective in my experience) but they will help you retain some heat at night. They are a bit expensive, but could give you some increase in heat retention. Be sure if you try these, to try and leave your pool pump off at night - they are most effective when the water is perfectly still - that allows the chemical to settle to the still surface to inhibit evaporation.