Bleach is useless on lawns. Organic matter inactivates the cleaning agent of it. It's why you have to clean up messes before you use bleach to clean up. So since a lawn is basically all organic matter, you won't do any good lol.
Diluting the areas where the puppy potentially pooped would be the way to deal with it in your yard. Given how long it lives in your yard, and pending the location - if you have neighbors who could possibly walk young animals/immunosuppressed ones nearby - it likely won't hurt to flood the yard a few times to try and dilute the particles down. If you do that a few times in the span of a few days, you should make it harder for an animal to pick it up. It needs 7 solid months of direct sunlight to kill it (which is why it can live a few years, depending on the location on your yard).
The virus sucks. It is a really well protected virus.
But as someone else pointed out, since your dog is older and likely properly vaccinated, you should be okay.
Titers are antibiotic titers. They take a blood sample and measure how many antibodies there are floating around for a specific disease. People will often do them to avoid needing additional vaccines, and to make sure the animal/person is properly protected if exposed to a potential disease since everyone is unique in their own way lol.