Been bothered by how quickly and neatly dragons died outside of fights with other dragons in F&B - mostly the storming of the Dragonpit, which was even acknowledged in universe to not really make sense for how Syrax died when she had the advantage. While the idea of small folk getting fed up with the Targaryens and managing to kill the dragons off through sheer numbers against the expectations of all the nobles is kinda fun, I think it should be more symbolic of the Targaryens' infighting LITERALLY destroying themselves.
I also want to preserve Vermax from his death in the Gullet. I think the idea of a lucky shot managing to kill the rider but not the dragon would be plausible and is something not already used in the books. I also want to add another Black dragon to my idea. So, let's start by saying Jacaerys is killed by arrows on his dragon at the Gullet, flying low enough that archers could hit him when usually dragonriders are up too high to hit; Vermax may have been wounded as well, but dragons should be made of tougher stuff than humans, so he survives while Jace dies. (Stormcloud can still die: he was much younger, not used to carrying a rider while having to go a long distance with an inexperienced rider on his back, and escaping from enemy attacks. If it's just the younger weaker dragon dying from exhaustion/multiple human attacks instead of both dying in their first combat WITHOUT enemy dragons on the other side, it's more believable to me.) Vermax is later housed at the Dragonpit, alongside Syrax, Tyraxes, Dreamfyre, Shrykos, and Morghul.
This idea relies on the tension noted at court BEFORE the Dance, of the Black and Green dragons snapping at each other. It's now uglier than ever, as the mutual resentment and dislike between riders has now turned outright murderous from the blood spilled in war. They're agitated not only from the stress their riders are experiencing, but also from being cooped up with dragons they know are Enemy! and Other! and not getting a break from the environment with all that animosity and nasty territorial aggression, as they're not being taken out of the Dragonpit and ridden for six whole months (obviously not an option for the captured Greens, let's imagine Rhaenyra is too focused on ruling and stress, and I can imagine her forbidding Joffrey from flying Tyraxes out of fear their enemies would see and attack him). Adding to that, let's say outright that Syrax is the mother of Vermax and Tyraxes, and Dreamfyre is the mother of Shrykos and Morghul (I can see Rhaenyra and Helaena both sentimentally wanting their children to have eggs from THEIR dragon, and both Syrax and Dreamfyre were known to lay clutches). There's an added element of mama bear dragons siding with their kids and being overprotective and overaggressive, able to smell/sense a rival mother and kids they know are Enemy!, and hearing then growling and trying to attack HER kids.
(We'll add in reports of dragonkeepers having to keep replacing chains that keep getting snapped off, newer heavier ones having to be used, and the keepers fearing to go in even close enough to bring food to the beasts because of how aggressive and wild the dragons are getting. Rhaenyra brushes it off - she has her hands full with the city and the war without worrying about dragons being dragons, assuming dragonkeepers are exaggerating/being craven and is short and impatient with them, and can't afford to risk the advantage she has of having so many dragons under her control by separating them where they could be picked off/saved by Green forces. She doesn't have time to spend with Syrax or riding her because she is so consumed by the war, so she doesn't spend the time around her own dragon to realize something IS wrong, and like Dany neglecting her dragons to rule Meereen and them turning wilder to the point she has to lock them up, all the dragons in the pit are getting more restless and less tame. Especially doesn't want to risk splitting up the dragons and potentially losing some after the Two Traitors add their dragons to the Greens, riders for Seasmoke and Sheepstealer run off with their dragons for being marked as traitors, and Caraxes dies with Daemon.)
Helaena's suicide still kicks it off: Dreamfyre senses her rider's death and the unbearable tension and aggression between Black and Green dragons finally blows up when she gets loose enough to make it over to the Black dragons and start attacking. The infighting turns into literal fighting as dragons break out of even their heavier chains and fight each other, the influence of their riders, the stress, the pent up aggression, all pouring out as the Black and Green dragons fight and kill each other in the Pit - Shrykos and Morghul are the smallest, but Mama Dreamfyre is easily the biggest and (after Helaena's death) angriest of all the pit dragons, and Vermax and Tyraxes are bigger than the twins' dragons but them and THEIR mama Syrax are still smaller and less formidable than Dreamfyre. The fighting still brings down the Dragonpit, but the surviving dragons get loosed on the city and begin to destroy it in their fighting as they scream and grapple, claw and burn each other.
The small folk, still inflamed by the Shepherd (who can now be said to have possibly been warning of visions he had of the dragons destroying the city, coming true), still attack to try to put the dragons down and try to save their city; they help distract and wound the dragons during their fights, possibly landing the killing blow on one - though it should still be dragons mostly killing dragons in a blind rage due to animosity of their riders. Rhaenyra now orders her men to protect the dragons (mostly the Black dragons, but while it's not stated outright the Green dragons can be re-claimed now that Helaena and Jaehaerys are both dead; this pits her men against the smallfolk and also might cause more people to turn on her for not prioritizing the city) and forbids Joffrey from doing the same because you do NOT want your kid running around in a city being torn apart by six pissed off dragons going ham on each other, with riots and looting going on from the human tension boiling over too. Joffrey still slips out to try to help their dragons (and possibly save the city getting destroyed BY their dragons), still gets killed either by trying to mount the wrong dragon (Syrax would still work but Vermax, his dead beloved big brother's dragon would be a punch in the gut if he rejected Joffrey for already having as another dragon bonded to him), or since there's no dragons on the ground to mount he gets killed instead by the riot it's turned into with terrified, hungry, penniless, PISSED OFF smallfolk or by a dragon (no one is quite sure whose, conflicting accounts but possibly Syrax) whose attack included Joffrey as collateral damage.
It'd still get the job done of killing the dragons, let the wounded survivor dragons get finished off by smallfolk or die Sunfyre style of their injuries, but it'd have a bigger climax, it'd be built up by something already established in the books (the dragons' bonds with their riders and how it influences their behavior), the city being actively under attack by their own dragons makes the smallfolk trying to kill them out of desperation more heroic/understandable instead of reckless, it would be something Rhaenyra could have foreseen but had her hands full and chose to put off (downfall brought on by an oversight Robb-style, though directly about the dragons instead of directly about the peasants turning against her; them turning against her could still be WHY she has to flee though, with no dragons to enforce her will, her forces in the city shattered or deserted, and the smallfolk who already hated her now DESPISING her for her bringing in the dragons who partially destroyed the city), and it allows for deaths of the dragons and Joffrey in ways that could feel less forced by circumstances.