In addition to the Clippers (formerly Braves), a couple other pro sports teams that figured it out and moved on without belaboring.1. NBA's Warriors, Philadelphia and San Francisco, 1947-1969
They recognized a "Warrior" didn't have to be a Native American, particularly associating Native Americans with persons willingly engaged in combat.
It took a little longer for them to realize they could even get away with the name without an anthropomorphic human mascot at all.
Unfortunately, it's harder for D.C.'s team to simply put up a silhouette of the Washington Monument and still call themselves something referencing the perceived skin color of a human being. Many joke that the R*dsk*ns could simply change their logo to a batch of mashed red potatoes and be done with it! Here's PETA's idea:
I'm down with that! Makes me good and hungry, too! 2. NBA's Blackhawks/Hawks, Tri-Cities (Moline, Illinois) 1949-1951
The owner of the Blackhawks didn't have any particular sense of compassion over the use of Native American imagery. Instead, he recognized the historical reference (to the Blackhawk War, waged largely in the Tri-Cities area of Illinois and Iowa) didn't hold any relevance to the Milwaukee area he was moving his team to.
Absent that forethought, we could have eventually had a team in Atlanta dealing with a similar issue as the R*dsk*ns do now. In the alleged "City Too Busy To Hate," in the state that's home base for the Trail of Tears, the MLB's Braves are racially insensitive enough for one town to have to deal with.
Thankfully, a simple name transition decades before saved us the trouble of arguing with the "But we've always done it this way!" faux-traditionalists.
I've suggested this before, but to me the slam-dunk modification would be to rename D.C.'s football team the Washington Pigskins. Pigskin is a well-known name for the obloid object they play with. Using the name "Skins" for short lives on. They can continue their traditions with the "Hogs" and the "Hogettes". Hey, it would tick off PETA, but in some people's books, that's a plus, too!
What's most telling, particularly with the Warriors, is fandom remains high decades after the name/logo change. No one is logically pining for a return to imagery of yesteryear, and holding the changes against those teams. It's a lesson Washington's owner would do well to learn so he can let go of his truest fear$.