I have made a lot of bamboo backed bows. For your bamboo, level the belly of the slat, draw your bow pattern out on the slat as perfectly as you can, cut out your pattern and sand it to the point that every edge is 1/6" thick, this will naturally taper your bamboo slat from the handle to the tip.
1 1/4" is plenty for a BBO, if you go out to the suggested 1 1/2" you will have a lot of bambo and almost no belly wood.
Use your bambo bow blank pattern to draw the bow profile on your slat, make the osage core slightly larger than your bamboo pattern because the bamboo may shift a little during glue up.
Don't worry about grain runout on your core slat, your bambo wild hold things together. I used my twisted tight ring osage for most of my BBOs, with over 50 of these bows in the books none of them ever failed because of grain runout.
Here is an example.
Taper core slat by making a 1/2" mark on the sides at the end of fade and drop that mark 1/16" every 6" until you get to 1/4" and carry that measurement all the way to the tips. I file to the line on the edges but leave the belly slightly rounded to give me more wood to tiller. Where the side measurement is 1/4" I leave the overall thickness about 1/2".