Bikram / Yoga or Pilates ?

I wouldn't call Bikram Yoga relaxing, though there is some definite mental benefit from it.

It's really hard if you go at it.  I've been doing it 1-2 times/week for about 2 years and have seen some benefits.  I don't know if they directly translate to triathlon but I think it helps keep my 45 YO body from falling apart (more than it is already, that is  ).

Yoga over pilates, for sure.  Good for stretching and strengthening.   I have only gone to one pilates instructor, so maybe there are other types.

I also am not a bikram fan, personally.  I get sweaty enough on my own.  I'm prefer Vinyasa.

I did tons of Bikrams, but it get's boring.  I guess that's where the mental aspect comes in.

Gave it up for heated Ashtanga/Vinyasa Flow/Power yoga.  If you could only do ONE physical thing for some reason, then this would be it.  Cardio, strength, flexibility, core...  It's amazing.

However, it's not something you should do in addition to your regular training, per say.  It's a workout and should probably replace one of your other workouts.  For example, it might be tough to do a 60 minute power yoga class (75 for Bikrams if I recall) in the morning then have a long run session later that day.  OR, add a Bikrams/Power to your day off.  That should be your recovery/rest day and those forms are NOT rest.

 There so many forms of yoga, and several of the "Hatha" forms are more for relaxation/meditation/stretching that could be incorporated with tri training and even done on your rest days, IMO.