It does not matter what your Software Engineer does at any particular time. Software Engineer is a professional job exempt from overtime rules for a reason: the thing that matters is whether he produces an appropriate amount of quality work.
Do you have any idea whether his work gets done? It doesn't read like you do. That's the question you should ask as you decide whether to keep or fire him.
If you want this IRS exempt employee to work for you by the hour rather than by the task or by the mission, you have that option. It's called "consultant" and you'll pay twice as much or more -per hour-.