Have you actually checked into the requirements for programming jobs in your area? I’m sure there are programming jobs that do not require a CS degree but, in my experience, the labs only hire people programmers with CS degrees. My experience is on the technical side, not the business side, but even most technicians and administrative assistants have a Bachelors degree in something.
Timoshenko was a mechanics person who laid the foundation for subjects like statics and mechanics of materials that you will see in Civil Engineering, not a CS person. I think the contrast between the two subject areas was @turbo93’s point, and I fully agree. I also have a BSCE (worked in the consulting business many years ago) and advanced engineering degrees but I have been doing engineering software development for several years now,
I don’t think the extra time to get a CS minor or CS degree really buys you anything for a career in CE. While some programming experience would be valuable, it would be the basic skills not the things that are taught in upper level CS classes. IMO, the most valuable programming experiences for BS level civil engineering would be in Matlab, excel, and/or python.