Schrödinger cat (SC) states are important resources for continuous-variable quantum computing and quantum metrology. In our previous work [1], we proposed a probabilistic protocol to generate bright squeezed SC states using degenerate spontaneous parametric down-conversion (SPDC) with pump depletion, followed by projective measurement of the pump mode. Here, we present a general theoretical description of SPDC with pump depletion, introduce an efficient numerical method for simulating its dynamics, and develop a practical version of our earlier protocol.