I agree with talking to your longarmer before purchasing batting. You also need to find out how big she wants your backing to be; some require larger excess backing than others. You also need to talk to him/her about the pattern, as he/she may have quilted other log cabins and have some suggestions. Not to mention that cost will vary depending on whether the longarmer freearms, uses a robot, or something in-between. Custom quilting patterns will cost more than pantographs, for instance.
I do not use bamboo batting because most of it is created with processing that uses highly toxic chemicals (toxic to both humans and the environment).
For softness and warmth, wool batting is preferred by many for bed quilts. Cotton is traditional and another good choice.
For baby quilts, cotton is a traditional choice that is also very good; however, if you are a new quilter accustomed to modern-day comforters, you may think that cotton is too flat for the look you are expecting. If you want more of a puffy look, Hobbs 80/20 is primarily cotton but with a little more loft than 100% cotton. If you don't mind synthetic, Hobbs PolyDown is a polyester that provides a lot of loft and softness and dries quickly in the dryer (which some mothers prefer to the slower drying time of cotton).