Where would he take this shower? In the bathroom off the murder room? And risk getting blood on himself on the way out? He was smart enuff to know not to do that. Maybe wash his hands so he wouldn't contaminate his dress shirt, tie, pants for his meeting. Got to do that now -- or in his hotel room if he had time. Which, which? Anyway, grab some clean clothes and & socks from his closet, being careful not to contaminate them with blood. Further, they had no heat, so how much warmer would it be? Plus he didn't have time.
Look at himself -- did he need to wash his hair? His face? He wisely planned to change clothes, shoes, etc., after he had exited the house. Maybe if he'd had the time, he might have staged a break-in, but no time. All he needed to do was change clothes, shoes, maybe discard his socks if they were bloody, throw both pairs of shoes in a trash bag along with his gloves and clothes, check his arms for blood, wash what he had to, back in the car, remembering to watch his speed, and GTH outta Dodge. Or something close to that, IMO.