I bought a cheap large Avenir pannier to carry a spare Ezee battery but as supplied it wasn't up to the job, just sagging shapelessly under the weight. I solved the problem by making a box to fit tightly inside the pannier with aluminium brackets to hook over the carrier and foam padding inside. The inner box is made from a lightweight pvc material called FoamX, normally used for signs, with a plywood base, although plywood would have been perfectly suitable throughout. The modified pannier can be seen
here, opposite the water container. It could be mounted lower with longer brackets to allow room on the rack for a dog basket.
Incidentally Redalpha, is your dog basket an enclosed cage or is your dog very well behaved needing no restraint and is the dog happy to ride on the bike? I ask as a prospective dog owner, when I eventually retire.