Ember vs Lava Plume HINT: it isn't even close

For simplicity's sake, these comparisons use attempts at doing straight damage without healing, use of support moves, etc. Jarold's HP is also ignored and it is assumed he lasts long enough to deliver the final blow against the pokemon in question.

The pokemon chosen are reflective of 3 situations. First, Laplace is a pokemon with multiple hard hitting moves and isn't hindered by movepool. Second, Babbage is a pokemon with one good move that can't be used turn after turn. Finally, Turing is a pokemon with a weakness to fire. Sierpinski would have been included, but fuck I am not calculating the average damage for fury attack sorry.

TLDR:
Against Laplace, using ember results in 226 HP loss. Lava plume results in 78.
Against Babbage, using ember results in 102 HP loss. Lava plume results in 58.
Against Turing, using ember results in 67 HP loss. Lava plume results in 10. (note, this is a OHKO, so the only damage taken is the HP cost of the feat)

VS LAPLACE

Laplace is assumed NOT to have def and sp. def combat stages, for simplicity, although this is NOT the usual case.
Jarold vs Laplace (ember)

Jarold uses Ember
Costs 3 HP, deals average of 27 damage before accuracy is applied, deals 23 after accuracy. Subtract 14 sp. def for 9 damage a turn. On average this takes 7 turns to take down Laplace, but since the HP for the last turn is so low, we'll use 6 turns for simplicity's sake.

This costs Jarold 18 HP from using ember.
Jarold takes 5 rounds worth of damage from Laplace (the 6th round, he KOs and Laplace does not get a move)

The optimal damaging move order is air slash, thunderbolt, air slash, thundershock, air slash.

Thunderbolt (1 time) will do an average of 61 before accuracy, 58 after.
Air Slash (3 times) will do an average of 47 before accuracy, 42 after.
Thundershock (1 time) will do an average of 25 before accuracy, 24 after.

Jarold takes 208 damage from these moves, for a total of 226 HP lost in this encounter.

Jarold vs Laplace (lava plume)

Jarold uses Lava Plume
Costs 10 HP, deals average of 49 damage before accuracy is applied, deals 42 after accuracy. Subtract 14 sp. def for 28 damage a turn. On average, this takes 2 turns to take down Laplace.

This costs Jarold 20 HP from using lava plume.
Jarold takes 1 round worth of damage from Laplace (the 2nd round, he KOs and Laplace does not get a move)

The optimal damaging move is thunderbolt.

Thunderbolt (1 time) will do an average of 61 before accuracy, 58 after.

Jarold takes 58 damage from these moves, for a total of 78 HP lost in this encounter.

Notes: Additional complications include - double team makes the fact of needing additional hits even worse by compounding the evasion issue. Confuse ray and thunder wave also put Jarold at a disadvantage if used while he is spamming ember, especially because paralysis becomes harder to resist over time. Additional chances to use Air Slash also give a higher chance Jarold will be flinched. This is not accounted for in this simulation.

VS BABBAGE

Babbage is assumed NOT to have def and sp. def combat stages, for simplicity, although this is NOT the usual case.
Jarold vs Babbage (ember)
Jarold uses Ember
Costs 3 HP, deals average of 27 damage before accuracy is applied, deals 23 after accuracy. Subtract 13 sp. def for 10 damage a turn. On average, this takes 6 turns to take down Babbage, but since that last turn is used to take down 1 HP, we'll use 5 turns for simplicity's sake.

This costs Jarold 15 HP from using ember.
Jarold takes 4 rounds of damage from Babbage (the 5th round, he KOs and Babbage does not get a move)

The optimal damaging move order is psybeam, tackle, psybeam, tackle.

Psybeam (2 times) will do an average of 40 before accuracy, 38 after.
Tackle (2 times) will do an average of 14 before accuracy, 13 after.

Jarold takes 102 damage from these moves, for a total of 102 HP lost in this encounter.

Jarold uses Lava Plume
Costs 10 HP, deals average of 49 damage before accuracy is applied, deals 42 after accuracy. Subtract 13 sp. def for 29 damage a turn. On average, this takes 2 turns to take down Babbage.

This costs Jarold 20 HP from using lava plume.
Jarold takes 1 round of damage from Babbage (the 2nd round, he KOs and Babbage does not get a move)

The optimal damaging move is psybeam.

Psybeam (1 time) will do an average of 40 before accuracy, 38 after.

Jarold takes 38 damage from these moves, for a total of 58 HP lost total in this encounter.

Notes: Babbage has conversion2. This makes ember exceedingly bad in an extended use situation. Babbage also has recover. Psybeam has a chance of confusion.

VS TURING

Jarold vs Turing (ember)
Jarold uses Ember

Costs 3 HP, deals average of 27 damage before accuracy is applied, deals 23 after accuracy. Subtract 12 sp. def for 11 damage a turn multiplied by 2 for weakness to 22 damage. On average, this takes 3 turns to take down Turing, but two since Turing will use take-down. However, this is done in such a way that Turing will optimally choose to use take-down on his last turn, self-KOing.

This costs Jarold 6 HP from using ember.
Jarold takes 2 rounds of damage from Turing (the 2nd round, Turing self-KOs from take-down)

The optimal damaging move order is metal claw, take-down.

Metal claw (1 time) will do an average of 30 damage before accuracy, 27 after.
Take-down (1 time) will do an average of 42 damage before accuracy, 34 after. Turing recoils for 9 damage, just enough to KO him after a second ember.

Jarold takes 61 damage from these moves, for a total of 67 HP lost in this encounter.

Jarold vs Turing (lava plume)
Jarold uses Lava Plume

Costs 10 HP, deals average of 49 damage before accuracy is applied, deals 42 after accuracy. Subtract 12 sp. def for 30 damage a turn multiplied by 2 for weakness to 60 damage. This OHKOs Turing.

Jarold spends 10 HP in this encounter. Turing gets no action.