Hi,
Have same dilemma, my son's passport is first expiring in Dec. Hope to travel soon to Europe (mother is a european so should work even if child only US passport).
Passport isn't expired, but its within the 6 months. is this rule really enforced?
First reach out to the embassy where your mother lives to grant you a permit - (it is not so easy to obtain, you might have to say that your mother is old and sick etc. My family is also from Europe and several family members were denied exceptions to enter. They are racist's against USA citizens...)
then see if you can also get an approval for your son.
Regarding close to expiration passport IDK, I would assume if it's a minor traveling with parents it's likely to be allowed, but that's my assumption. Take along a copy of his birth certificate for a better chance.