Well depends on what you mean by "long time". Battlemaster has the head start, though Hex helps warlock stay close.
Assuming...
3 Action surge = +3/6 attacks
Agreed
archery style = +1/2 attack
No idea how you are getting this
Or this
frighten = Web (including pushing into it)
Each enemy targeted by a spell that can prevent them from acting that provides a repeated save/attempt to end the condition is worth about 2 uses of frighten. The math on this is below.
Each use of frighten has a 60% chance to cause an enemy to lose a turn (you'll only use it when that's a possibility). Thus, 2x frightens will be 1.2 turns denied.
A use of a 60% reoccuring save spell will on average prevent 0.6 + 0.6*0.6 + 0.6*0.6*0.6 = 1.18 turns (in a 3 round fight) or 0.6 + 0.6*0.6 + 0.6*0.6*0.6 + 0.6*0.6*0.6*0.6 (in a 4 round fight) = 1.31.
As you can see, a single enemy targeted by a repeating save spell that denies actions will deny about as many as 2x frightens.
Then, I'd estimate a single level 2 spell can likely target 2 enemies (tasha's laughter is guaranteed to do so with a good effect).
In which case 1x web = 4x frightens. Except you need a web for each short rest and since you've given this fighter 2 based on the number of actions surges then that's 3x web's required to match up with 4 superiority dice all going to frighten over 2 short rests.
Not sure how you are getting this
1 Sorcerer Point = 1 attack (Seeking Spell or Quicken)
Agreed.
You can’t trade Pact Magic for SP.
I forget about that restriction.
Warlock2/Sorcerer 2: 5 SP = 5 rerolls +2 Hex = 7 attacks
BM: 4 frightened and 6 attacks
W2/S 3: 11 SP = 2 Web + 7 attacks
BM: 4 frightened + 9 attack
W2/S4 4: 14 SP = 3 Web + 6 rerolls
BM 6: 10 frightened + 9 attacks
W2/S5 5: 23 SP = 3 Web + 17 rerolls
BM 7: 10 frightened + 9 attacks
There's way to much you are not accounting for in your methodology.
1. Each of your EB attacks is worth less than the battlemasters. This cascades to things like action surge and seeking/quicken. In which case either Action Surge should be worth more than 1 attack or seeking/quicken should be worth less than 1.
2. One obvious miss on your part is that you aren't accounting for the +18 damage provided by maneuver dice per short rest. Or for 2 short rests that's +54 damage. That alone is worth nearly 11 EB attacks.
3. It doesn't appear you are accounting for archery style being applied to action surge either, though that's probably only a bit more than 0.6 attacks.
4. Same thing for the +1 mod as for archery style.
And that's, IMO, where it pulls ahead.
That's assuming your alone. Get any ally to help, (grapple, Web, Push, Booming Blade) even once per encounter, and Cloud of Daggers will pull ahead.
Those things still have to connect and you have to compare the value of what your ally would have done without cloud of daggers to that.
I'd say Web (+ Push) is comparable. Locks their movment and they have disadvantage.
The effect is comparable enough, actually a bit better.
If you need 3rd level slots, then 11 sounds more like the break point.
I think by the time the battlemaster gets 5 manuever dice (level 7) then you either need an extra level 2 slot or some level 3 slots mixed in to break even on the control aspects. I mean you can load up on control options, so it's possible you may can target weak saves or hit more enemies than 2/3, etc, all of which may provide additional values to the spells in the comparison, but none of that is nearly a sure enough thing to add it directly in.
Feel free to confirm those numbers with more detailed math.
I'll go back and do a full pass on them.
And it takes all the BM resources too.
Yes. My point was that there wouldn't be much left for things like shield and etc.
Also, you got 2 Warlock slots for Shield or whatever.
In my initial assessment I was wrongly converting those to sorcery points. Since you can't do that then I'd probably need to add another sorcerer level. But yes, at that point you would have the resources to do everything and shield.