I think you are doing it the only way you can. No way to avoid the internal face. Just draw a new line across the junction of the vertical and horizontal faces to separate the hidden face the delete the unneeded edges.
After the second Push/Pull operation, select all of the geometry. Right click on it and choose Intersect Faces>With Selection. Then erase the edges on the faces.
Frankly, I would probably not draw the bottom part of the bracket until the upper part is completed and avoid creating any internal faces or coplanar edges to deal with at all.