After playing it extensively with multiweapon TMG+Rifle with the TMG as the base weapon, and TMG+Sword with TMG as the base weapon, I have noted a number of strange occurrences.
After starting the counter for Chain Trigger with a charge TMG photon art, attacks from the sword or rifle on the multiweapon would either cause the counter to disappear, cause Chain Trigger to immediately end and go into cooldown (this usually happened only with rifle and sword photon arts that you could charge), or cause the counter to freeze on whatever count it was at. When the counter would disappear, sometimes Chain Trigger would still be functioning, as TMG attacks would still make the noise of the counter going up, and a charged TMG photon art would do more damage from detonating it.
Additionally, sometimes Sword and Rifle photon arts that can be charged up could begin Chain Trigger, though nearly always with the counter being invisible. This did not happen very often, but it enough that I am certain it was occurring.
There were no instances of a Sword or Rifle charged photon art being able to detonate Chain Trigger, but given how random the other behaviors would occur it might too be possible.
Hopefully these are all fixed by launch so Gunner players can enjoy the use of multiweapons without messing up the ability to use their main class ability. Thank you for your time.