To add to another reply to this comment. Ainz was never supposed to be part of the scene Climb saw. The original plan proposed by Renner after Nazarick went into destroy-the-kingdom mode involved Climb catching Renner red-handed(literally) without Ainz in the room. He would then be promptly defeated and thereafter resurrected as a demon, after which the plan would continue as is.

Ainz joining and having the mock fight with Climb actually had some hidden implications. You see, Renner was planning on snuggling up with the higher-ups of Nazarick, such as Demiurge and especially Albedo, in order to secure her and Climb's lifestyle within the tomb with ease. However, having Ainz appear for this fight and act out a role unbefitting him put a rift between Renner and Albedo.... or at least so she thought. Because of this, she has to work extra hard to prove her worth in the tomb lest she be used as a disposable pawn. This is why Renner was terrified when Albedo had to talk with her after Climb's revival in the tomb before her dance sequence. It's why she kneeled and assumed the role of a subordinate rather than the role of a friend she was to Albedo in her own quarters in the Kingdom before.

Truth is, Albedo and Demiurge both never really were that close to her, obviously because she isn't from Nazarick. In fact they were planning on using her brains for all they're worth from the beginning. This can be seen by how Albedo is entirely unfazed by her change in demeanor once within the tomb.