AI-assisted authoring requires users to frequently exchange user-generated and AI-generated content between a primary workspace and a Large Language Model (LLM) assistant. Such repetitive actions can fragment attention and compete for limited display space. While Augmented Reality (AR) has been proposed as a way to extend desktop workspaces with virtual displays, just adding a spatial display area does not address the repeated attention shifts, input-focus changes, and content-transfer steps required in LLM-assisted workflows. This creates a gap in understanding whether AR second screens support such workflows or introduce cross-reality transition costs that limit their effectiveness. To investigate this gap, we designed an AR-Second-Screen system around a unified keyboard-and-mouse interaction model and evaluated it against Single-Screen and Dual-Screen desktop setups. Results show that Dual-Screen remained more efficient than AR-Second-Screen, with significantly lower task completion time and switching cost. AR-Second-Screen also produced higher task and LLM idle ratios, indicating re-engagement costs when users crossed between physical and virtual workspaces. At the same time, participants rated AR positively and preferred it significantly over Single-Screen, with no significant difference in preference compared to Dual-Screen. These findings show that spatially extending AI-assisted workflows into AR can be appealing, yet effective cross-reality interfaces must minimize transition costs rather than just simply adding a virtual display space.