If you have GC or your are citizen, your child can't use 旅行證. You and your child are foreigner in China. your child needs to get a visa from China embassy.
旅行證 is for american born people but their parents donot have GC yet. So the child actually has double citizenship, the child can live in china as long as he can.
I have no info for GC holder or U.S. citizen in China (I donot know). But you should be able to live longer than a year, and you just need to renew your 暫住證 when it expires.