I done traveling with my daughter using a few options :
1. Waist pack for camera -there are models from Lowepro, Tamrac, Tatonka, etc. I use the Lowepro one now. Had a Tatonka one that I really liked. Doubles as a sling type camera bag in crowded places where you want the bag in the front. This frees up the shoulder for a standard backpack/daybag.It also distributes the weight of the camera gear from waist down, reliving the shoulders and back of its weight.
Baby stuff will go into a separate daypack.
2. Shoulder bag (Saffrotto - Domke F3 copy). Very large bag. Fits loads of camera stuff with separate compartment space left for baby clothes and diapers. Wet/water/food goes into either the side pockets or a daypack.
One more thing. Don't look at the baby stroller as a hinderence. It can be make life much more comfortable. Baby can sleep, drink milk, rest, change diaper on it. It also serves as 'pack mule' to hang your camera bag and daypack. You only have the few seconds inconvenience loading it up a transport vehicle.
Bring one unless you are very sure you are always on unpaved roads.
Good luck, enjoy your trip and post up the photos when you are back
It is perfectly fine to bring one for Phuket.