Actually Drake, I think the order went like Murasa/Ichirin > Shou.
Though I think Ichirin might be the first one.
The reason is that, at first Byakuren appeared out of nowhere as a youthful monk, implying that she might not have a temple(or it's currently abandoned), and only by then that she started gaining her fame among the people. So people heard of her reputations and then she was sent to subdue Murasa with other monks, that are likely not from her temple.
Ichirin is quite a wild card, since she can be before, between, or after Murasa and Shou. But I think it's weird to get her after Shou. Since she's a youkai-turned human, it would make sense for Byakuren to get her first, as the crew that has the easiest time communicating with the humans.
After a while, Byakuren either created a temple for the Youkai, or used the old temple of Bishamonten. Then she appointed Shou to take care of the temple, along with the help from Nazrin to monitor her. The temple became flocked with Youkai, and some of them might've been seen by the humans, thus leads to their suspicions of her.
Though I kinda agree with you too. It makes sense for her to leave her temple for a journey when she had Shou to look after it.
But they didn't mention how long it takes for her to finally be trusted by Bishamonten. So it could be within those few decades after Murasa's case. Which is why I said she might be last.
Well, they didn't really give any specific time, so it's quite difficult to clarify it.
And yeah, now that I think about it. It was never mentioned that Murasa and Ichirin followed Byakuren after their respective meeting with her. Or is there?