I'm not sure if you are interested in trying homeopathy with Max, but I'll make a suggestion or 2 here, in case you are.
My 3 yo has respiratory issues related to (mostly fall) allergies. He gets very wet in his lungs and sometimes ends up with an upper respiratory infection because of it. Because of this, remedies that are useful with 'wet lungs' are the ones I'm becoming most familiar with.
One of the ones that helps him is Spongia -- which is what I use when his lungs sound 'spongey'.
In my book "The Homeopathic Childbirth Manual" by Betty Idarius (who's both a homeopath and a midwife) another of my favorite respiratory remedies is listed under 'newborn care.'
Antimonium Tartaricum: "For respiratory distress from fluid in the respiratory tract. The baby's throat and lungs are full of mucus, which you will hear rattling. The respiration is accelerated. This is the main remedy for meconium aspiration with the above symptoms. In extreme cases there will be severe rattling, gasping, breathlessness, paleness..."
Camphora: "is the remedy to use if Antimonium Tartaricum seems indicated but fails to act.
"The dose for a newborn is the same as for an adult. To prevent possible gagging or spiting out of the remedy, cruse a dose between 2 spoons and place the resulting powder into the baby's mouth. You can also dissolve the remedy in water, which is especially convenient if you need to give a frequently repeated dose..."
Of course if you have access to a homeopath, that would be the ideal situation, so that he/she could evaluate Max's entire situation (including any other residual birth trauma) and recommend the best remedy. Failing that as an option, though, I believe that the Antimonium might bring him some relief.