Context. Contrary to expectations from scenarios of black hole growth driven by galaxy interactions and mergers, dual active galactic nuclei (AGN) with kiloparsec separations are rarely observed and are very difficult to identify, in particular at high redshifts (i.e. z > 2). Aims. Focussing on the recently discovered dual AGN system LBQS 0302−0019 at z = 3.29, we seek to identify further group members in its environment and to understand their formation history through deep high-angular-resolution imaging. Methods. We present deep Hubble Space Telescope (HST) Wide-field Camera 3 near-infrared imaging of LBQS 0302−0019. In combination with ground-based VLT/HAWK-I imaging, we infer accurate sizes, colours, ages, and stellar masses of companion galaxies. Results. We clearly detect four companion objects close to LBQS 0302−0019 that also have faint signatures in the ground-based images. We constrain light-weighted ages and masses for the two most prominent companions, Jil1 and Jil2, to t⋆ = 252−109+222 Myr with log(M⋆/[M⊙]) = 11.2−0.1+0.3 and t⋆ = 19−14+74 Myr with log(M⋆/[M⊙]) = 9.4−0.4+0.9, respectively. The HST data also show that the obscured AGN, previously identified by strong nebular He II emission, is associated with the young massive companion Jil2. Because very massive stars of the starburst cannot be solely responsible for the He II emission, we strengthen our initial conclusion that Jil2 has been hosting an AGN. Conclusions. If the young starburst of Jil2 had been accompanied by sustained black hole growth, Jil2 may have contributed He II-ionising flux to create the large He II Lyα proximity zone around LBQS 0302−0019. Hence, the duration of the current luminous AGN episode of LBQS 0302−0019 may have been overestimated.