I concur with the successive rounding rules described above. I'll throw my two cents out there and suggest maybe making zatgun attacks require a contested dexterity save on the part of the target. This is in line with the effect of the weapon: no real way to defend except for special armor or more practically to dodge/roll out the way. In much the same way that some spells have DC's in 5e I would think that energy weapons could benefit from having a similar mechanic (energy weapons are really just scifi spells no?) that could be penalized at certain ranges. For example, the default DC of zatgun could be 8 + dex modifier + any proficiencies - some penalty for being at far range.
Furthermore, requiring a constitution check or save every round by the stunned target could be a good way to determine the duration of the stun effect. Perhaps the default behavior is: On a failed dexterity save the target is stunned for a maximum of 1 minute (10 rounds). At the start of a stunned targets turn they make a constitution save. On a success the target is no longer stunned. Insert above mentioned successive round rules here. If a stunned target is stunned by a zatgun again (not consecutively) it counts as a failed constitution save for their next turn. (i.e. a player can spend their action to keep they enemy stunned but in doing so may not be able to do anything else. This could keep zatgun spamming to a minimum...)
I also agree with idea that death by zat should really just drop the pc to zero hp and trigger standard death saves.