> Clearly, $A(t)$ is the analytic signal of $v(t)$.


How so ? This is simply a conversion from cartesian to polar co-ordinates. Every complex signal can be written this way regardless of whether it's analytic or not. 

> But the Hilbert transform sort of overpredicts the imaginary part,

It does not. It calculates the correct imaginary part for an analytical signal. It's probably not what you expected but that's an issue with your model assumptions, not with the transform itself

> so that Hilbert transform fails

The transform doesn't fail. It does what it is defined to do. Your result looks perfectly correct. 

> So, I am wondering if there is a better technique that can construct the complex A(t) from the given real-valued measurement v(t)?

First you need to define what $A(t)$ actually is. Your original definition is doesn't help since it applies to every complex signal on the planet. I think you want something like an "envelope" but you need to specify clearly what that is and what it isn't in the context of your specific application and requirements.