So Matt pointed out that the section in Sihler where he (Sihler, not Matt) goes over syllabic laryngeals may shed some light on why a laryngeal which stands in between two consonants fails to show the triple reflex in Latin. Apparently this is the reason: it's a syllabic laryngeal, which develops differently in Latin as well as Sanskrit, for that matter. (CHC > Lat. CaC, Skt. CiC)
I don't know if I like this explanation. I can't quite put my finger on it, but something seems fishy. I wonder whether Hittite shows any reflexes of these syllabic laryngeals. Off to Kloekhorst!