Can Bringer of Dreams and Nightmares use "To Dream a Thousand Deaths" to generate Fear/Push Invaders that are protected by the Heart-Tree Incarna?

Can Bringer of Dreams and Nightmares use "To Dream a Thousand Deaths" to generate Fear/Push Invaders that are protected by the Heart-Tree Incarna?

Yes. Since "To Dream a Thousand Deaths" ensures that Bringer of Dreams and Nightmares doesn't even attempt to deal Damage or destroy any Invaders/Dahan/Beasts with its Powers, there's no destruction for the Heart-Tree Incarna to prevent.
(This could have been ruled either way – "To Dream a Thousand Deaths" is structured in a way that makes clean ruling difficult. The ruling is partly for thematic/story reasons and partly so that players who already played it this way won't feel like they cheated.)