The general approach is okay but it does not have to be anything like this complicated. Protein A is not a fantastic binder to IgE according to the paper, but clearly someone has had at least some success, and assuming you want to take this approach you could modify the process to make it easier.
First you could just buy protein A-Sepharose resin (you don't need glass beads and it is quite unusual to see that). Second, you can pack simple disposable columns and use gravity feed instead of pump systems. The main issue for you is binding of other subclasses and the paper discusses options for applying protein G in the process.
I would suggest a different approach if you can purchase an IgE-specific secondary antibody. You could conjugate that that to a bead (CNBr sepharose would be the easiest approach as there is no difficult chemistry involved - just wash the beds and add the antibody in the correct buffer). Then you can do immunoaffinity purification and the presence of IgG in the sample becomes irrelevant - it will just flow through the column. After washing you can then elute IgE with pH 2.3 buffer and neutralize the eluted material quickly with Tris buffer.