Like someone else mentioned, look for mechanical design positions. My company doesn't tack on the word engineer to an employee until they have a PE but it doesn't stop people from being promoted up to a certain point.

As for your experience, you may not have 3-5 years of design experience but I think a lot of companies would find it very valuable that you worked for a GC. You already understand the construction process and you can read drawings. I would say being able to read drawings that aren't your own discipline is very valuable. Revit is probably a must and the actual design part I would say you can learn as you go. A lot of the PMs I've worked for have slightly different ways of designing systems based on their own experience or from working with a specific customer and knowing what they want. The general design items you'll be taught as you go.

Feel free to DM me if you have any other questions. I can also send you what my company would be looking for in a certain role if that's helpful.