The hybrid barium titanate (BaTiO3 or BTO) – silicon nitride (Si3N4 or SiN) platform integrated on silicon has been established as a promising candidate for implementing photonic integrated circuits with unique features in terms of high modulation speeds and low transmission loss. However, despite the high Pockels effect in BTO, switching device footprints are relatively large, which could compromise their use in applications with large scalability requirements, such as neuromorphic computing hardware. To address this limitation, we propose the integration of vanadium dioxide (VO2) in the SiN/BTO platform to enable ultra-compact amplitude switching devices offering a scalar multiplication functionality with multilevel operation. More concretely, an electronically reprogrammable switching device with a 5-bit amplitude encoding capability and an insertion loss of only 0.5 dB is developed. The proposed device is built with a 9-µm-long VO2/SiN/BTO waveguide structure integrated with an efficient microheater using a transparent conducting oxide. Such a vital building block would offer significant potential for developing more complex photonic integrated circuits, including dot-product or matrix-vector multiplication engines. The combination of high speed, low loss, and reduced footprint makes the proposed enhanced platform an attractive solution for application in scalable and energy-efficient neuromorphic computing hardware.