The foam sounds like a good idea. I was thinking you could tape the strands together down a couple-few inches from the tips, just enough to wear the business end of the wires would contact the batter section of the cajon. The tape would keep them from jingling but still allow them to make a "snare" sound on the box. I suspect the jingling is from the separate wires percussing against one another.
BTW, if the snares don't work out, consider some sort of exterior pocket or something that would hold an existing shaker. Based on more the past
17 years of playing acoustic shows with singer/songwriters, I find certain shakers, rectangular, or pouch shakers filled with BB's or seeds, etc., sound much more like a snare drum than snare wires on anything other than the resonant head of a snare drum. :-)Using an exterior shaker also give you the opportunity to customize/change the sound by simply attaching a different shaker.
Back in the djembe fad of the noughties (2000s), I kept a velcro pad glued to the head of my djembe and spots of velcro stuck on just about every shaker device I owned. This allowed for a bunch of variation in sounds.