Blistering Invective deals persistent damage:
Your words deal 2d6 persistent fire damage, and the target must attempt a Will save.
Energy Fusion adds a different damage type to a spell that deals energy damage:
If the next action you use is to Cast a Spell that deals acid, cold, electricity, fire, or sonic damage, select a non-cantrip spell in your spell repertoire that deals a different type of energy damage from that list, and expend an additional spell slot of the same level as this secondary spell.
Do these work together?
Best Answer
No, but...
No, as I explain in my answer to the question Medix2 linked in comments, applying the same (seemingly accurate) ruling suggests that the combination does not work.
Specifically, persistent damage is "dealt by" a Condition, not the character that inflicted the Condition. By extension, the spell is not "dealing" damage in the sense that Energy Fusion implies; it would be different if blistering invective had a duration over which it did damage instead of being the Condition persistent damage.
But... this is something a GM could easily rule the other way on. It will increase the effectiveness of Energy Fusion a bit (I believe the "expected" duration on persistent damage is approximately three ticks) at the 'cost' of using less up-front damage. A 4th level blistering invective does 4d6 damage to 3 creatures whereas a 4th level fireball (ye olde go-to for spell damage comparison) does 8d6 to everything in a 20ft radius. It takes a second round for BI to catch up in damage assuming fireball only hit 3 targets.
However, GM's should take care as there are other spells that trigger persistent damage, and it's possible that, depending on group composition and enemy types/density/etc., adding sometimes 2-4+ times the bonus damage from Energy Fusion (in addition to the non-damage effects of such spells) could cause issues with spells hitting well above their weight.