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The function weierstrass_p() of EllipticCurve returns the Laurent expansion of $\wp$ at the origin, hence evaluating it only gives a reasonable approximation near it.
Nothing comes to my mind to do this kind of numerical evaluation straightforwardly in Sage. If you have access to Maple, you could give a shot at the NumGFun package, part of the AlgoLib library http://algo.inria.fr/libraries/. It has support for the numerical evaluation of functions satisfying linear differential equations with polynomial coefficients.
The author of NumGFun gave a talk on it at Sage Days 49 http://www.marc.mezzarobba.net/#expose-sd49. As you can read in the slides, nothing of it is already in Sage, but there are plans for the close future. See, for example, http://trac.sagemath.org/ticket/14996, which will add functionality similar to what you need.
