I'm not sure if there is such a specific fanfiction. Fanfiction can cover a wide variety of scenarios, but this particular one might be a bit rare or not in line with the general tone of the original work. You could try searching on some major fanfiction websites to see if it exists.
Honestly, I haven't come across a fanfiction with that exact plot. Fanfiction often takes different directions based on the creativity of the fans. Some might focus on the characters' battles, their growth, or their friendships. While the idea of Ichigo and Unohana in such a situation is possible in the realm of fanfiction, it's not a common theme that I've seen so far. You could check dedicated fanfiction platforms and use relevant search terms to look for it.
There may or may not be. Fanfiction is a creative space where fans can explore all kinds of relationships and plotlines. However, this concept seems rather unorthodox considering the nature of the characters in the original series. It's also possible that if it does exist, it might be a very niche piece of work that not many people are aware of.