If you've been following this thread, you'll have noted a lot of head scratching in regards to different Helmholtz resonator neck profiles (ie sharp corners vs flared vs chamfered etc). I found time to do a few more tests, and think there might be something to this.
To recap, resonators with smoothly flared necks (eg bottles) seem to be acoustically reactive while producing minimal candle deflection (caused by the phenomenon of "vortex shedding"). Resonators with sharper hole edges (eg MDF boxes) appear to show little acoustic reactivity, but produce large candle deflections at resonance.
The holes in the MDF boxes, in addition to having sharp edges, can also be characterized as "long and narrow" in the sense that the neck length is three times the neck diameter for the configuration required to target my particular room mode.
In place of the MDF boxes, I started using capped ABS pipes with similar volumes to the MDF constructions and a machined plastic insert for the neck. This is more controllable and easier to ensure air tightness.
First, I mated an ABS pipe with a neck insert that matches the MDF constructions: 1/4" diameter x 3/4" long, with sharp hole edges. This behaves very much like the MDF boxes, in that there is very little acoustic reactivity (compared with the backside of the tube as control) but a very large candle flame deflection (ie vortex shedding).
Chamfering the edges of the 3/4" deep hole insert led to a slightly larger departure from the "control" frequency response, but it was still nothing like the bottles.
I then removed the plastic insert, and considered the hole through the ABS pipe cap as the resonator neck. This neck, with a diameter of 3/8" and depth of around 3/16", could be considered "shallow and wide" compared with the "long and thin" insert (which mimics the MDF hole). While this inevitably changed the resonant frequency, it also had the effect of greatly increasing the acoustic reactivity and also of greatly diminishing the candle flame deflection at resonance.
I then added a modest chamfer to the shallow hole in the MDF cap, and observed an even larger increase in the acoustic reactivity (keeping in mind that this changed the resonant frequency again, but only a bit). In this configuration, the candle flame deflection was completely eliminated.
Here are some photos of the neck insert (after chamfering) and the ABS hole with and without a chamfer. In all cases, a chamfer was applied to both the inside and outside ends of the hole.
And here are the frequency response curves for the various HR apertures:
What I've been calling "acoustic reactivity" is manifested as the bipolar excursions in the orange and cyan curves around 130-150Hz. The increase in reactivity when going from a long, narrow hole with sharp edges (red curve) to a shallow, wide hole with chamfered edges (cyan curve) is very evident. Another observation is that, with the chamfered 3/8" hole, it is possible to excite a pure tone by blowing across the opening (something I've been unable to do with long/narrow/sharp-edged necks).
So at this point, a reasonable working theory seems to be that long narrow holes with sharp edges are prone to vortex shedding, which impairs the acoustic reactivity of a Helmholtz resonator. MDF panels in 3/4" thickness have been a common material for DIY attempts that I've read about, and when targeting lower frequencies you will inevitably end up with the (apparently) undesirable long/narrow/sharp-edged hole. The issue is related to non-linear turbulence effects that are not modelled in popular HR calculators, which will happily predict excellent absorption properties in what I am christening the "candle snuffer" regime.
My next steps will be to adapt the ABS tube geometry to get into the frequency range of interest, and then get back into damping.
Thanks to Endorka for the suggestion of experimenting with hole inserts!