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We present a full 3D numerical evolution code to study neutron stars in massive scalar-tensor theories. The code is embedded in the Einstein Toolkit framework and its implementation constitutes a modified version of the Baumgarte-Shapiro-Shibata-Nakamura formalism with an additional nonminimally coupled scalar field. The approach we follow preserves the standard hydrodynamic evolution for matter fields, allowing eventually for a straightforward inclusion of more microphysical effects and better flexibility. Using this code, we examine the gravitational collapse of rapidly rotating, scalarized neutron stars to a black hole by exploring the influence of the scalar field on the dynamical features of the process and on the gravitational-wave emission. We find that for the configurations studied in this work, there is an observational degeneracy in the tensorial gravitational-wave emission between collapsing scalarized stars and their counterparts in general relativity. However, this degeneracy can be broken through the emission of scalar radiation, which carries an energy of <inline-formula><mml:math><mml:mo>∼</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>M</mml:mi><mml:mo>⊙</mml:mo></mml:msub><mml:msup><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></inline-formula>. This is orders of magnitude higher than the quadrupolar emission (<inline-formula><mml:math><mml:mo>∼</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>7</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>M</mml:mi><mml:mo>⊙</mml:mo></mml:msub><mml:msup><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></inline-formula>) and might be used as an observational probe of modified gravity. We also find that rapid rotation can enhance this signal, since fast rotating stars can sustain larger scalar field amplitudes.