Even within a single transfer switch (either automatic or manual) which switches the neutral, many include a special mechanical construction such that the neutral is not interrupted until a fraction after all ungrounded leads are opened, and the neutral is then closed to the alternate power source before any of the ungrounded conductors are closed. Note that simply using a three-phase three contact TS (without explicit provision for the neutral) will NOT give you that extra protection that a single phase three wire transfer switch could include.
The alternative, to completely prevent even a temporary open neutral condition, is to use an overlapping contact switch that parallels the two sources momentarily during the transfer (with all of the specialized protective gear that requires.)
The possibilities of the additional relay having mismatched timing characteristics or just plain failing lead to a high risk of damage to equipment.