Transfer factors for Jacquet-Mao's metaplectic fundamental lemma

In an earlier paper we proved Jacquet-Mao’s metaplectic fundamental lemma which is the identity between two orbital integrals (one is defined on the space of symmetric matrices and another one is defined on the $2$-fold cover of the general linear group) corrected by a transfer factor. But we restricted to the case where the relevant representative is a diagonal matrix. Now, we show that we can extend this result for the more general relevant representative. Our proof is based on the concept of Shalika germs for certain Kloosterman integrals.