I don’t know what the courts were thinking, but here is a guess:
The state says that non-employees in the chamber are a security threat. The courts almost always defer to the executive on this sort of assessment.
The state could probably argue that it does not have the resources to hire a Muslim chaplain. Remember, it doesn’t even have the resources to hire the guards it needs.
So how to avoid the problem of religious preference? Either
(a) ban the Christian chaplain from the chamber, or
(b) point out that a state-employed chaplain is expected to provide cross-faith spiritual support to people of all faiths, not just their own. Their personal faith is irrelevant, just like the personal faith of every other employee is irrelevant.
But regardless of whether you end up with (a) or (b), Ray’s imam still wouldn’t make it into the chamber. Therefore, his request is not particularly likely to succeed. But at this point in time, he needs to be very likely to succeed to get a stay of execution. Therefore, no stay of execution.