Dog fighter is kind of a strange term, since the definition whether it involves close-range manoeuvring or is just aerial combat is a bit difficult. I will, for the following, assume it to be aerial combat.
The fighter with more energy is still going to die if it tries to turn away from it's enemy to climb again because you can not climb fast enough to get out of range in the 3 seconds it will take the enemy to turn and fire on your unprotected back.
This is wrong.
Basically, the tactic looks like this: You dive in straight towards the enemy (well, to a point a bit forward in his course), then open fire once he's in range and climb out on the same course. Only then do you turn and dive in again. And again. And again. You do not need a good manoeuvrability to correct for his evasive manoeuvres since you are far enough away (see triangle formulas for that).
So, the enemy has three possibilities if he detects you (note the if) [Note: For the following I am assuming you to come in from behind him; it works just the same for the sides or from the front but changes the numbers a bit]:
- Continue on course, climbing or diving: You still have the advantage of speed and more energy, meaning you catch up to him and can zoom away again. If he tries climbing after you, he will lose speed (which you do too but can afford as you have a higher speed from your dive) and will risk stalling. This is the only time he has an opportunity to engage you. I'll come back to that in a bit.
- Go to the sides: You can still correct your course and get him. If not - do another pass later. He cannot shoot at you.
- Reverse course: Still able to correct. If not - again, other pass. He cannot shoot at you.
Engagement time: The engagement time depends on the speed of approach (subtract the two speed vectors) and the guns ranges, plus the time needed to point the plane in the right direction. Of course, only a fraction of that will hit depending on pilot skill, but let's just assume they're the same fractions and therefore are not important. Let's assume a speed difference of 50m/s (180km/h) and effective weapons ranges of 200m.
If you dive towards the enemy, you have four seconds to pump him full of lead. The enemy, in turn, will have four seconds of his own to attack back - but since you are then already climbing out of the dive away from him, he needs to correct his course to gain an attack opportunity on you. Four seconds may seem much - but if you have to react to a plane zooming to your left, right, above, below or wherever, tilt yours to the right angles and then fire with an acceptable chance of hit, that's difficult. And that's only if the enemy didn't kill you during his burst.
Therefore, manoeuvrability, while important, is by far not the only important characteristic. [1]
I am still against building a twin-engined fighter for the present, since our range allows us to operate effectively without one for now. The advantages of twin-engined fighters is armament and range plus a higher engine output [Note: subjective. If I'm wrong, correct me.] We pay for this with a higher cost per fighter and less manoeuvrability. For now, I believe the exchange warrants a single-engined one.
[1] You may note that the above basically requires you to have an altitude/speed advantage. For two fighter groups on the same altitude and speed, the one with more engine power per weight and higher climb rate can force such an engagement, though, as it can outclimb the other. The advantage of twin-engined.
The point is though your enemies odds in any 1 single patrol circuit are higher that he'll get through if he only has a 5 minute window of being caught per circuit, equal over all but high speed does not make any actual difference for patrol.
Let us assume for now a single line of a thousand kilometres being patrolled by 50 fighter As and fighter Bs. Fighter A has double the speed of fighter B, 250km/h. This means that all of Fighter As are 1000km/50fighters = 20km/fighter apart. This means a time of a bit under five minutes until you see the next one.
Fighter Bs, now, are the same distance away. This means a bit under two and a half minutes until the next fighter gets to the same place.
This does not include the effect of ground maintenance, refuelling and so on (a plane isn't in flight 24/7), but there the twin-engined has the range advantage meaning more time in the air.