I remember reading somewhere in AIDA documentation that contrary to common belief, the urge to breath, is not an urge to inhale, but an urge to exhale CO2.
So, the exhale thing is a sigh that your CO2 i building up and you're diaphragm is starting to contract gently, you just don't feel it (the same way that most of the time you don't feel your heartbeats).
I also had this issue when doing DNF last year, exhaling some air makes you feel less the urge to breath (urge to exhale), but as explained in freediving courses, letting out air, means you'll lose pretious O2 and the O2 partial pressure will go down, and you make risk earlier Hypoxia, and in worse case LMC and BO.
I sometimes used a trick, when it's unbearable, like letting out air in my cheeks an then back to my lungs, but it only tricks my mind for a few douzens seconds.