We calculate the odd-parity, radiative (ℓ≥2) parts of the metric perturbation in Lorenz gauge caused by a small compact object in eccentric orbit about a Schwarzschild black hole. The Lorenz gauge solution is found via gauge transformation from a corresponding one in Regge-Wheeler gauge. Like the Regge-Wheeler gauge solution itself, the gauge generator is computed in the frequency domain and transferred to the time domain. The wave equation for the gauge generator has a source with a compact, moving delta-function term and a discontinuous non-compact term. The former term allows the method of extended homogeneous solutions to be applied (which circumvents the Gibbs phenomenon). The latter has required the development of new means to use frequency domain methods and yet be able to transfer to the time domain while avoiding Gibbs problems. Two new methods are developed to achieve this: a partial annihilator method and a method of extended particular solutions. We detail these methods and show their application in calculating the odd-parity gauge generator and Lorenz gauge metric perturbations. A subsequent paper will apply these methods to the harder task of computing the even-parity parts of the gauge generator.