With SCS you need a conic formulation so you need to use the ExponentialCone.
You see in the doc that (x, y, z) is in the cone iff y exp(x / y) <= z and y > 0.
So for instance exp(a) <= b is rewritten as [a, 1, b] in ExponentialCone().
It’s not clear to me how to reformulate your model into a conic program though.
It does not even clear that the problem is convex.
For instance, in 0.5 <= a * exp(b) + c * exp(d), for the problem to be convex, you would need a * exp(b) to be concave in a and b while it is convex in the direction b.
If you replace @objective with @NLobjective and @constraint by @NLconstraint, you can use a nonlinear solver like Ipopt or NLopt.