As a piano teacher, my blanket rule is to not start before year 2. Obviously there are children who are ready younger, but since you can't know each individual child before they start lessons, it is impossible to tell who those will be. Practice is hard at any age, but especially when they are little and don't understand why it needs to be done.
The main reason for my blanket rule is that the drop out rate pre year 2 is so high, simply because the children are so young. This can lead to children believing they are unmusical, whereas they were probably just not developmentally/ physically ready. Good teaching will help, but if a child has not touched a piano between lessons, then they will have to go back over the same ground in subsequent lessons rather than moving forward. Obviously over a number of weeks lack of progress becomes demoralising, and frustrating for parents paying for lessons.
Specific advice for the OP, I wouldn't get a smaller keyboard; playing on different size keys in lessons and at home will add confusion, and can over a long period of time lead to poor keyboard geography. If your DS can't physically play some parts of his music, the teacher should be showing him how to move his hands to facilitate his playing - I had one student who had lost a finger, but she still learnt to play. If he really doesn't want to practice, try asking him to play one of his favourite tunes from what he has already learned instead - better to play than not. Does the teacher give out practice charts? My littlies get a sticker if they have four smiley faces on their chart in a week.
If it doesn't improve then I would take a break till he is older, and ask for a different tutor book when starting again, to avoid negative connotations of 'I couldn't do this bit'.