I definitely agree with Split Testing as every subscriber base is different. This is evidenced by the spike in website traffic your client is seeing on Mondays. Monday is a quieter traffic day for many websites.
Most of us don't sit by our inbox waiting for the next email newsletter to arrive, so I think you'll be more than safe sending on different days without upsetting people
You are far more likely to annoy people and have them unsubscribe if your content is poor. Focus on content and appeal to your audience and they will be happy whenever you decide to send.