After a bit more playing around, fixed a few of the more obvious bugs, improved the parameter naming, and added the individual plot components as dashed lines on the plot. A version 2 of the demonstration notebook is attached.
I'm actually surprised, it's quite stable and usable. I was able to do a more complicated 5-peak voigt fitting with relative ease. After achieving the desired fit, I was unable to pickle the fit object itself, but I was able to pickle the parameters object. Or the fit report could be used to record the results instead.

If there's enough interest to keep developing on this, I'm considering that perhaps the best thing to do would be to start my own repository to house the project. The main lmfit project is quite stable and I think would be cluttered by trying to include the development of this interface, which is likely to take a lot of updates to get into a stable, reliable form. Also, it could be nice to keep the extra dependencies separate from the core package (as it looks like the IPython stuff has already caused some headaches in the past).
Does anybody have thoughts on taking it in this direction, starting a separate lmfit_jupyter_ui repository? Perhaps it's time to deprecate the ui subpackage of lmfit and remove those dependencies from the main project?