Ahhh oh boy I love this topic bc all my short stories are love stories that focus on the relationship, but have some sort of conflict that helps move the character's relationship forward. Actually, this is one of the things I really like to experiment with (different types of antagonism/conflict), and like you, I sometimes struggle to hit that happy balance of relationship development and conflict. It's one of my goals with every story to get it right (or just about!).
So idk how much of this will help, bc I'm still learning too, but I will share examples using my own work. In my stories there is not always a real defined antagonist (like a masked villain running about), but there is, like @BobbyjoeX other types of conflict present.
For example, in my short story "Barry and the Mountain", the conflict is Man vs Nature and Man vs Self as Barry attempts to wrestle with his grief by essentially trying to conquer a very tall mountain. The mountain god is not so impressed with Barry's ill-conceived attempt, and his desire to educate/help Barry is what brings the characters together.
In the case of "Danica and the Dungeon Spawn", the characters work together to defeat the villain, and so the focus shifts to how that grows their relationship.
In my short story, "Head Scientist and Chief Minion of Hell", the antagonist is job pressure (the pressure Satan puts on his head scientist to create unmentionable horrors). This job pressure drives the head scientist and chief minion closer together as the minion rallies to support the head scientist.
In the above examples I feel like the antagonist is definitely present, and drives the story to start, but fades away in the middle to allow for character relationship development, until the conflict/antagonist is resolved/defeated in the end.
Sometimes I feel like I don't do a great job at balancing. "Kolya and the Harmony of Winter" really focuses on the enemy, and the love interest doesn't come in until the end. I personally feel like that difference is really noticeable.
However, it depends on what you're trying to accomplish. I think "Sabriel" by Garth Nix is an excellent example of a fantasy novel that very much focuses on conflict while the romance is nice but takes a backseat. In my series, however, the romance is supposed to be center stage. If your villain is really integral to the plot, and you want the audience to really relate to that villain, then maybe the balance would be tipped more in your villain's favor.