OK. I see you've added those values.

k=7.6675*10^14,   
η =6.34963*10^-12,   
α0 = 0,   
α1=1.26993*10^-11,   
α2=1.90489*10^-11,   
α3 =2.3282*10^-11,   
α4=2.64568*10^-11,   
P0 = 1.0,   
P2 =1/2 (-1 + 3 Cos[θ^2),   
P4 = 1/8 (3 - 30 Cos[θ]^2 + 35 Cos[θ]^4),   
Exp1= Exp[ i * η * Log[(Csc[θ]/2])^2] ],   
Exp2 = Exp[ i * η * Log[(Csc[[Pi]/2 - θ/2])^2] ]

I have noticed that I have forgotten the unit conversion. So I have put a 10^28 coefficient in front of y then the script works and the parameter output is {δ0->1.23497, δ2->0.381176, δ4->0.569623}. But this time another error appears : "The step size in the search has become less than the tolerance prescribed by the PrecisionGoal option, but the gradient il larger than the tolerance specified by the AccuracyGoal option. There is a possibility that the method has stalled at a point that isn not a local minimum."

Thank you Gianluca. It seems you are right. I will try one more free parameter as a coefficient in front of the function. I hope it will work.