What these have in common, and what differentiates them from all others, is that they take advantage of the shift of the CP towards the CLA at increased AOA.
Two fins are not necessary for a belly flopper. (Well maybe just "flopper" as there would be no "belly.) I just don't want a fin to break on landing. I suppose it might be interesting to fly a "flopper" with three or four fins to see what the descent looks like.
I think you do indeed need a "belly". I think any number of fins (within reason, after 10 probably diminishing returns!) would work, but I theeeeeenk they have to by eccentric. The "belly" is the side opposite the exaggerated opposite fin surface area.
My hypothesis is you put a third equal size fin on your current version on the opposite side equidistant from the other two and you will have a lawn dart.
What confounds me still on your rockets is that I think by "Cardboard Cutout" method of stability calculation (which is I think basically similar to your Center of Lateral Area) your two fin rockets are STILL stable, therefore by that method they should lawn dart. Your first statement of the shift of CP from Barrowman to CLA definitely works for relative SuperRocs, with large Length to Diameter Ratios, BECAUSE (if I understand it right) Barrowman essentially ignores the length of the body tube in the calculation of CP because it is negligible AT NEAR ZERO ANGLE OF ATTACK (again, @neil_w
please correct me if it am wrong.) The Cardboard Cutout/CLA DOES ACCOUNT for the exaggerated long body tube surface area in long length to diameter rockets, and this surface are is enough to make the rocket unstable but only "kicks in" when the rocket is thrown off of near zero angle of attack. Your rocket flown with a plugged motor has no such "event", so "something else" must be happening between stable vertical flight and eventual horizontal recovery.
SOMETHING must happen to the rocket for it to transition from rapid pointy-nose first flight to slow sideways horizontal descent. For HSR and BSR, that even has classically been the puff port throwing the rocket into non-negligible angle of attack. You have proven by your plugged motor that Belly Flop Recovery doesn't need it. I don't buy (I may be wrong) that most rockets fly pointy nose up continuously until they run out of kinetic energy, and then start to fall tail first. The reason I don't buy it is that in every onboard video camera I have seen (and many of them are on high power rockets with electronic recovery timed for apogee) unless ejection is early the rocket always seems to be horizontal or nose down at ejection (@Steve Shannon
have an opinion on this.) Here is Steve Eves' Saturn V (former world record largest rocket launch) video, it noses over before deployment, it doesn't start falling tail first. Jump to one minute.
So I still theeeeenk that with Belly Flop Recovery, the initiating even is the loss of forward velocity combined with the eccentric fin placement. I think any SYMMETRICAL placement of relatively normal thickness fins (@Flyfalcons
got away with using really thick symmetrical fins in his award winning Guitar Rocket, hey if you ain't cheatin' you ain't trying!), any SYMMETRICAL placement of normal thickness fins that is stable on boost will REMAIN stable on descent without some kick (although we will see how @Dotini
does with his next rendition, I think he is moving puff port to CG which I don't understand, and even if it works won't prove that "something" didn't happen. I don't have the guts to try HSR with a plugged motor.)
I still wonder if Belly Flop Recovery isn't essentially Back Slide with really poor forward velocity, and that MIGHT be BECAUSE your rockets aren't SuperRoc high length to diameter. I REEEEEEALLLLY miss having a nearby flying field, as I would love to play with this more myself.