He's got the bug. Oh yes.
Inspired by Kingpin's excellent image I see it's now possible to try to build on an important point I touched on earlier:
...you lost this fight most pointedly because you did not neutralise a perfectly in plane nose-to-nose arrangement...
In plane manoeuvring.If you look at that jet, and imagine another blue arrow which emerges from the centre of the aircraft and shoots out the left wingtip, and another shooting out the right wingwingtip, then there is your 3-9 line. Now if you stuck a virtual Rotring or Staedtler or Messerschmidt compass (they always seem to be German) in the centre of that jet right down one of those 3-9 lines and drew a blue circle, one that went through both the lift vector and the arrow coming out of the nose, you could draw a blue circle around the aircraft. So if you looked up out the cockpit you'd see a line going fore-aft and right down through your nose and back through your stab: the axis upon which your elevators act. When you pull back or bunt then your nose would travel on the circle. The 3-9 line is the axle for elevator turning.
If you do a competition-winning Immelmann at an airshow, then your blue circle slides on a great big piece of paper hung upright in the sky. A 2-dimensional surface upon which the manoeuvre is performed, a plane.
Plane: 1 a flat surface on which a straight line joining any two points on it would wholly lie.In the case we've described, all the points of our circle, and the arrow heads of both our lift vector (and the 'thrust' vector) and the centrepoint of the plane are points on this surface.
So with ACM there are at least two aircraft involved. In your fight with Hammerdown the moments before the HO, neither of you used much aileron. Your 3-9 lines were both level with the horizon throughout and because you'd been chasing him as 'gunsight on' as possible you were both in exactly the same plane. Your big pieces of paper were hanging in the sky as one. You were manoeuvring and travelling in exactly the same plane.
Now the consequence of the turn rate and radius is very clear he's only got to pull round enough go match your flightpath and the gunnery solution drops in his lap as a gift! Your response is to deny or complicate his gunnery solution considerably by making it a 3D problem rather than a 2D one. So as you saw him pulling up over the water, even by the halfway point of his move when he's vert you should have anticipated the danger and rolled 45 degrees to one side and pulled up*.
Like Morfiend said here would be an appropriate moment to take your gunsight OFF the opponent and fly for position. Since you've rolled by using your ailerons your plane of manoeuvring
has rolled with you and is now
NOT ALIGNED with that of your opponent. Even if he adapted and was able to put his guns in your direction it would be a momentarily and difficult shot, an eye-blink at best as you are traveling quickly across his nose. You can easily jink out of those. Now look at the possibilities of relative position and energy, you'd easily get above and he wouldn't have the energy to follow. Now you can choose a better position to put your gunsight back in his general direction and pressure him.
He can't run, he can't hide, he can't pressure you, he can't dictate the fight, he can only sit in his cockpit blubbing for his mama and quickly leafing through the speed index of his copy of ACM for Dummies - Quick Access Version wondering what happened.
So in summary the solution to being in a situation where you are losing a nose-to-nose arrangement is to get out of plane. To cut inside you when you are above corner speed is suggestive that he is slower and further suggests your follwing move: take the fight into the vert. Punish him if he follows you up and punish him if he doesn't.
I learned this by-the-way not from reading manuals or having sessions with Trainers but trying to turn nose-to-nose with those bloody hover-flap Vought Corsairs at the furbal lake about 10,000 times, back in the day when I thought bloodymindedness was the answer
I know it looks complicated from all the text, but if you plod though it section at a time, making diagrams for yourself or just imagining if you're good at that and doing the fighter pilot's hand thing you might reach an epiphany worthy of a Christmas present.

Good luck. Keep at it.
*Actually here you might reverse your roll as you cross and pull and really describe a 3D flightpath, but that complicates the basic point.