We　propose　a　protocol　to　realize　nonadiabatic　geometric　quantum　computation　of　small-amplitude　Schrödinger　cat　qubits　via　invariant-based　reverse　engineering.　We　consider　a　system　with　a　two-photon　driven　Kerr　nonlinearity,　which　can　generate　a　pair　of　dressed　even　and　odd　coherent　states　(i.e.,　Schrödinger　cat　states)　for　fault-tolerant　quantum　computations.　An　additional　coherent　field　is　applied　to　linearly　drive　a　cavity　mode,　to　induce　oscillations　between　dressed　cat　states.　By　designing　this　linear　drive　with　invariant-based　reverse　engineering,　we　show　how　to　implement　nonadiabatic　geometric　quantum　computation　with　cat　qubits.　The　performance　of　the　protocol　is　estimated　by　taking　into　account　the　influence　of　systematic　errors,　additive　white　Gaussian　noise,　1/f　noise,　and　decoherence　including　photon　loss　and　dephasing.　Numerical　results　demonstrate　that　our　protocol　is　robust　against　these　negative　factors.　Therefore,　this　protocol　may　provide　a　feasible　method　for　nonadiabatic　geometric　quantum　computation　in　bosonic　systems.　©　2022　authors.　Published　by　the　American　Physical　Society.　Published　by　the　American　Physical　Society　under　the　terms　of　the　Creative　Commons　Attribution　4.0　International　license.　Further　distribution　of　this　work　must　maintain　attribution　to　the　author(s)　and　the　published　article＇s　title,　journal　citation,　and　DOI.