The growing demand for medical isotopes and mobile neutron sources has motivated the development of compact and multifunctional accelerators. The China Institute of Atomic Energy has developed an ultra-compact superconducting cyclotron, CYCIAE-36A, capable of accelerating H₂⁺ and α particles and delivering proton and α beams for applications including ²¹¹At production, PET isotope supply, α-irradiation studies, and neutron imaging. The highly compact layout poses significant challenges for beam extraction. A dual extraction system combining electrostatic deflection and carbon-foil stripping is proposed under the constraints of superconducting magnets, RF cavities, and limited space. In this scheme, α particles are extracted using an electrostatic deflector, while H₂⁺ ions are extracted by stripping, enabling variable-energy proton beams in the opposite direction. Phase width control is critical for extraction efficiency. A fixed-phase selector was implemented in the central region to optimize α-beam extraction. Particle tracking simulations starting from the central region assume phase widths of 20° for α particles and 50° for H₂⁺ ions, yielding extraction efficiencies of 65% and 99%, respectively. In addition, the effects of 1st and 2nd harmonic magnetic field components introduced by the magnetic channels during precession extraction were analyzed, and corresponding compensation methods were developed to mitigate their impact on beam dynamics.