I'm not aware that TNG officially established "Constitution Class" for the TOS
Enterprise.

I've just compiled
plenty of evidence in the "Enterprise-A" tech thread that we might be looking at a lot of confusion here.
Bob
From TNG's
Relics:
(bottle and glass in hand, Scott walks carefully to the holodeck)
COMPUTER: Please enter programme.
SCOTT: The android at the bar said you could show me my old ship. Let me see it.
COMPUTER: Insufficient data. Please specify parameters.
SCOTT: The Enterprise. Show me the Bridge of the Enterprise, you chattering piece of
COMPUTER: There have been five Federation ships with that name. Please specify by registry number.
SCOTT: NCC One Seven Oh One. No bloody A, B, C, or D.
COMPUTER: Programme complete. Enter when ready.
[NCC1701 Bridge]
(Complete with sound effect, they did a great job of recreating it for us. The viewscreen has the ubiquitous orange planet on it. Scott goes to his old station and pours a drink.)
SCOTT: Here's to you, lads.
PICARD: I hope I'm not interrupting. I was just coming off duty and I wanted to see how you were doing.
SCOTT: Not at all, not at all. Have a drink with me, Captain.
PICARD: Thank you.
SCOTT: I don't know what it is, exactly, but I would be real careful. It's real
(Picard knocks it back in one)
PICARD: Aldebaran whiskey. Who do you think gave it to Guinan?
SCOTT: Ah.
PICARD:
Constitution class.
SCOTT: Aye. You're familiar with them?
PICARD: There's one in the Fleet museum, but then of course,
this is your Enterprise?
SCOTT: I actually served on two. This was the first. She was also the first ship I ever served on as Chief Engineer. You know, I served aboard eleven ships. Freighters, cruisers, starships, but this is the only one I think of. The only one I miss.
According to the official TMP blueprints "the new design increases phaser power by channeling direct power from the warp-drive system."
I really don't know if this is much of a help, but it seems to suggest that like the "redesigned impulse engines" phaser power is drawn from the "linear inter-mix chamber" as well.
Bob
Don't forget the line from Star Trek II. After Reliant gets smacked around by the Enterprise, she has to withdraw and can't return fire. Why? Because "they damaged the photon controls, and the warp drive."
This bit of dialog meshes nicely with Decker's comment:
[FONT=Arial]
DECKER: Sir, the Enterprise redesign increases phaser power by channeling it through the main engines. When they went into anti-matter imbalance, the phasers were automatically cut off.[/FONT]
So, the power is increased by going through the main engines, and when warp drive is out, phasers cannot fire.