As someone who brings long-ago physics major studies to the work on
instruments, resonant circuits in electronics have inductance,
capacitance and resistance elements.

In acoustics, including the resonant spill valve oscillations we're
talking about here, the analogues are mass, stiffness and resistance.
Resistance in acoustical circuits is exemplified by things like the
felt and window screens over the vent that have been mentioned.

Greater mass (inductance) _reduces_ the resonant frequency (slows the
oscillation); greater stiffness (capacitance) _increases_ the resonant
frequency) and resistance broadens the resonant peak -- it makes the
resonant frequency less definite or reduces the "Q" of the circuit.

Springs and masses can both increase the pressure, depending how they
are used.  (Organ regulator bellows often have both.)  Resistive loads
should have minimal effect on pressure, but by broadening the peak
resonance, they reduce the likelihood of oscillations starting, and
damp them out when they do start.

A small mass attached to the leather flap might slow the oscillations
off the resonant frequency of, say, the cavity where air is vibrating.
Gently closing the flap might eliminate the stimulus that starts the
oscillations, and felt around openings and the window screen across
the opening might destroy the resonance by slowing the air flow at the
mouth of the cavity.  Just some thoughts on the relevant theory.

Good luck; it will be interesting to hear what works!
 
Rick Hunter