A sealed cavity full of light weighs more than an empty one. So consider a cavity consisting of parallel mirrors, with a pair of beams bouncing between the mirrors. The question is whether that weight will change if we arrange for the beams to completely cancel one another at every point along their length?
You're not being very careful about your terminology. Instead of weight I think you're really talking about the total mass-energy of the sealed cavity. If you want to discuss weight the it would be better to think of that as the force on the total mass-energy in a gravitational field.
Thinking about it in terms of mass-energy makes the question much easier. Then it's obvious that the massless photons still add energy to the system (already acknowledged in your question) and (hopefully!) it's obvious that no constructive or destructive interference or absorption of the photons will change the total mass energy.
That is, due to the conservation of energy, nothing that happens to the photons inside of the sealed cavity will change the total mass-energy of the system.