Although the cults of Yelm, Orlanth, and Yelmalio are rivals, they are perfectly capable of working together. The cults are Neutral towards each other – not Hostile or Enemies.
At the same time, Orlanth and Yelm, or Orlanth and Yelmalio, are not Friends. The relationships between these cults are unstable. Sometimes they can work well together for a common cause or against a common enemy (usually Darkness or Chaos). Often their latent rivalries result in conflict and even war.
In the end, the biggest source of conflict is that both are Ruling Gods. Yelm acknowledges no other as his ruler, and nor does Orlanth. Yelmalio inherits Yelm’s position – the Little Sun has no ruler but the Sun (and that ruler is distant or even gone).
The relationship between Orlanth and the Red Goddess is very different. Those cults are mutually hostile, even enemies.
Note that Yelmalio is hostile towards the Red Goddess. He is neutral towards the Seven Mothers.
Is it the same in the Lunar Heartlands, or are the Antirius/Yelmalio cultists there hostile to the Red Goddess? They likely are, with priests raising concerns about the Red Goddess’ Chaos association, while at the same time carrying out their contracts and agreements with the Seven Mothers and Red Emperor (and dutifully obeying the Yelm Imperator leaders).