The inscriptions are all names, except for 5 characters written around the perimeter of the disk, which say
祈武運長久 (pray for good luck in battle), a very common message on these types of flags.
There is nothing else on the flag that gives any hard clues as to whom the flag was given. There are a lot of names with the surname Tachimatsu, so it could have been given to someone from this family.
I would say there is a non-zero chance that any surviving family members would not want the flag. Sometimes these war mementos just bring back painful or perhaps uncomfortable memories, and instead of bringing closure it just brings up awkward feelings. Enough time has passed so that its probably a higher chance that they would welcome the return of such an item, but some families prefer to let war memories die out. (I do think its highly admirable to at least make the attempt to return it... I just throw out the contrarian info so you have a more complete picture).