My daughter’s boyfriend is in construction management. His degree (undergrad) is construction management but it was part of the engineering dept. D is civil engineer and they had a lot of classes together. He also has an MBA. He makes slightly more than her but he has the MBA and an extra 2 years working. She’ll someday be a PE and he won’t.
I think in years to come she’ll stay with one employer and probably make less, and he’ll take risks and jump around for different projects. Different comfort levels with risk.
Construction managers tend to have higher salaries (and higher bonuses especially when working for developers), but work longer hours. A normal week for me nowadays is 55 hours (6:30am-5:30pm), with some weekends as well. Sometimes, I have shorter days, sometimes I have longer days. It can vary quite a bit.
Stick with CE or ME and take as minor in construction management. This will give you the most flexibility in a 40 year career and overall highest earning potential.