TWWG is spot on.
BIBE is brutally hot along the river and the high desert during July and August. Absolutely brutal. We have been to BIBE several times in July and August. We limit our desert and river visits to the early morning hours...and are usually off those trails by 11AM at the latest as the heat really builds after that.
The high Chisos are another story - very comfortable, even in August. Daytime highs can reach the upper 80's or low 90's, but it does cool down into the 70's or 60's, so it is comfortable. As a result, the hiking, backpacking and camping in the Chisos are very comfortable, even in the summer. The chance of rainfall picks up during the summer. We have had a couple of trips out there during the summer when we encountered rain, but it was over quickly.
One of the main advantages to visiting BIBE in the summer is that the park is very empty. So, you would find good availability on the campsites in the basin, I would think...