Berries require an acid soil, pH 6 or so, and your Joshua grows in alkine soil, pH 7.5 or so. The soil will not be good for either plant. The roots of the tree will continue to grow outward as it grows so I suggest you look for native berries which there are several available.

Unless you want to loose the Joshua Tree, keep all watering away from the base and outwards to a distance of at least 25ft. A 4ft tree is very old and not used to excessive water. Joshua's are very prone to rot. Try and not disturb the soil in that 25 ft area either. The tree is probably 100s of years old.

Berries will do better if grown in a more moist environment.
Blueberries need very acidic soil. Raspberry and Blackberry a little less so. Boysenberries do real well in SoCal.