The bilayer iridate Sr₃Ir₂O₇ is a c-axis collinear antiferromagnet, held there by a giant interlayer pseudodipolar anisotropy, whereas single-layer Sr₂IrO₄ cants in the ab plane. We show from first principles that biaxial compression of a few percent (ε_c≈-2.4%) flops the easy axis of Sr₃Ir₂O₇ into the plane. A magnetic model Hamiltonian built from Wannier functions with no fitted parameter—reproducing the giant magnon gap of the bulk, so far known only from fits to experiment—identifies the mechanism. Compression collapses the interlayer exchange channel, whose straight Ir–O--Ir path weakens as the bent in-plane path strengthens. Hund's exchange sets the scale of the anisotropy and, beyond J/U≈0.15, removes the collinear state altogether. The flop is not a rigid rotation—the ordered moments of the two states cross at ε_c---and it carries a stark fingerprint, in that the giant easy-axis magnon gap collapses to a gapless, Goldstone-like spectrum. Compressively strained films thus sit on a metamagnetic phase boundary ending in a zero-temperature bicritical point, a charge-neutral handle on spin–orbit-entangled order.