Why doesn't he just try to adopt her properly???
Honestly, it feels like anime and manga sometimes over complicates things for no reason...
If they explain the situation to the authorities, he could apply to be her actual guardian. It might take time, but at least he won't be known as a pervert.....
Yuri has a living mother and possibly living father, so under Japanese law what the gov't would first do is drag either (or both) of them to the Ward Office to instruct them to pick her back up. In the case of proven child abandonment the gov't would then take her under protection (usually, but not always, in an orphanage). The best case scenario would be to have her stay with a volunteer foster family approved and vetted by the government. In fact, given her age, usually this stay would only last until she enters high school, at which point the Ward system would have allowed her to start living alone while the gov't collected her living expenses from her living parents. She would also have been allowed to work to earn her own keep, at least partially. In case both parents are unable to support her financially, the gov't would supplement her income from Ward system. Adoption would not be necessary.
Even in the case of a biological uncle offering to adopt, the parents must sign a bunch of paperwork to agree to adoption, then other living members of the family, starting from grandparents and aunts, would be considered before the uncle is even allowed to proceed. The uncle would only be considered if no alternative family member exists or all potential guardians have declined. Even so, the government would likely attempt to offer financial help for raising the child before moving to the next-in-line. There would be an assessment of the uncle's family (it is STRONGLY EXPECTED that he'd be married). There would be a period of at least one year of observation where the uncle('s family) would be required to demonstrate the ability to take care of a minor while the minor spends minimal time in the expected new home, and another year of probation/adjustment where the child would be allowed to start living with the uncle('s family) during which rather intrusive inspections by social workers would be the norm. A single male living alone would usually fail the first step by default, even if he's the biological uncle. It is basically impossible for an older, unmarried, unrelated male to adopt her.