Or a mei tai, either something with an adjustable base (if bubs prefers legs out, eg: catbird baby, or the pikkolo which is a buckle version if you are really adverse to straps), or babyhawk, or the Storchenwiege half buckle carrier is excellent for newbies. Actually, that would tick all of your boxes, it is adjustable for newborn up to toddler, it is made of wrap quality cotton so is supportive yet very cool, it has a good headrest for newborns. Personally I'd recommend something with good adjustability because you want to keep a reflux baby properly upright, so a half buckle would be able to be adjusted better. For the same reason I'd suggest trying a close carrier before committing to one, as it is stretch fabric you may find that bubs slumps without frequent adjustment.