We study the singularity category of the Brieskorn-Pham singularity R associated with a Geigle-Lenzing projective space X of weight quadruple. We introduce the notion of 2-extension bundles on X, and then establish a correspondence between 2-extension bundles and a certain important class of Cohen-Macaulay R-modules studied by Herschend-Iyama-Minamoto-Oppermann. Furthermore, we construct a tilting object in the stable category of arithmetically Cohen-Macaulay bundles on X consisting of 2-extension bundles, whose endomorphism algebra is a 4-fold tensor product of certain Nakayama algebras. We also investigate the Picard group action on 2-extension bundles and obtain an explicit formula for the orbit number, which gives a positive answer to a higher version of an open question raised by Kussin-Lenzing-Meltzer. This is a joint work with Shiquan Ruan and Weikang Weng.