As I understand it, the point of the experiments with the shifting screen was to compare the brilliancy of colour when monochromatic light exposed a part of the plate all by itself, and when it was combined with other monochromatic wavelengths. Brilliancy and luminosity were OK with two colours, but dropped noticeably with three. The difference between a monochromatic source and a fairly wideband source with the same centre wavelength was dramatic.
I (well, Ives and common sense) would suggest using a three narrow passband filters and exposing the same plate with each filter in turn. If you have three light sources (small flashes, for example) you can use the same filters on them and make a single exposure with all three lights on at once.
You can think of the lamellae as being like a comb. If you have two combs of different spacings and hold them next to each other you can see a combined pattern of light and dark along the comb - in acoustics it would be a beat, but in visual sciences it is usually called a moiré pattern. As you add more frequencies it gets harder and harder to see the gaps, and you get more and more overlaid tines. With a continuous spectrum you end up with mostly mush. Ives has curves of the density vs. depth profile for continuous spectra, and sure enough, they are weakly modulated and not very fringe-like at all.
Fourier theory gives you an out: you can always separate out the individual components if you have long enough combs, but the point with Lippmann emulsions is that you don't have enough depth of the lamella structures to distinguish between a large number of close wavelengths. Worse, the coherence problem, and scattering within the emulsion, mean that you can never get the required depth.
So the answer is to use a single red, a single green and a single blue wavelength, rather than a continuous spectrum. You can do this with single wavelength sources (LEDs are cheapest and safest), or with narrow band filters on the lens and wideband illumination.
So what if your exposure times are measured in days. It worked for Ed Weston :-)