I have done plenty of bargello quilts using fat quarters , you just need more of the same strip sets. In fact I prefer not to work with the full width on very large strip sets it get tough to keep them all going straight and it gets heavy. I have done quilts as large as queen size bargello with fat quarters.
If you have ever seen the pattern twisted bargello I did that quilt with primarily fat 1/4's. The fabrics that were full width I cut in half to make all the strips sets the same lenght.
See photo