If she sees that she cannot control him and that he will do as he pleases, no matter her demands, she would not think twice about taking his bond from one of the women who holds that power over him. Despite Alanna's inability to "handle" him, Egwene would believe that she is far more able to use the bond to force him to her will. And there is no way in hell Rand would allow her to bond him. If she wants it, she'll have to take it from one who already holds him.

Additionally, if she sees the world's leaders falling in behind him, rather than siding with her, holding his bond would effectively get them in line - after all, everyone knows that a Warder is nothing more than an Aes Sedai lapdog. Plus, Egwene has been set up as flat out godlike in that she can snap someone's mind with her iron will (Mesaana), so it would not surprise me in the least if she can force Rand to her will using the bond. It will disgust me, but it would not surprise me.