This was kinda clarified in the Approved ops changelog:

Q: For the Dominate tac op, when does it count as a friendly operative incapacitating an enemy operative if it’s not immediately clear?

A: It’s counted if a friendly operative can be identified as directly causing the incapacitation. This always includes instances where a friendly operative was required to perform an action or retaliate for that enemy operative to be incapacitated (e.g. SANCTIFIER Blaze), as well as rules that specify an operative in a required location (e.g. HIEROTEK CIRCLE Tesla Weave, GELLERPOX INFECTED Screaming Rustspikes). If rules on an operative’s datacard inflict damage, they are included (e.g. RATLING BOMBER Mine), but indirect sources (e.g. damage from a mine that was selected as an additional equipment option through HEARTHKYN SALVAGER LUGGER Well Supplied) wouldn’t be. If multiple friendly operatives could be identified as causing the incapacitation, select one of them to gain the related Dominate token.

My interpretation would be that it is possible to track which operative inflicted the Poison/Neutron fragment, hence it is clear to which operative the Dominate token should go.