Probably not; I don't think Katniss could necessarily be argued to be living "happily" with Peeta at the end of the story but it does read as if Katniss is peaceful, resigned, and accepting of her life, relationships, and experiences. Gale wanted to be active in the changes and have a more involved role in what was going on post-war and in the future so that didn't really fit with who Katniss was and became through the Games and by Mockingjay. The Katniss at the beginning of the story might have eventually had a future with Gale but that Katniss exists primarily as a contrast and memory to the Katniss she becomes and that Katniss ending up with Gale would have been a years-down-the-line Katniss that would still likely have a different worldview than Gale and that we wouldn't have gotten to read and that there might have been no story to. Katniss is still similar to her previous self but with more trauma, loss, and instability so no, she probably would not have been as happy (or as peaceful, settled, etc.) with Gale in the story as written.