I'd like to be able to measure the activity of $\beta$-galactosidase in living cells with simple optical (maybe fluorescence) microscopy. Ideally I'd like to do a minimum of genetic engineering, and use this assay with strains I already have (that have a WT lactose system), i.e. a fluorescent lactose-mimic would be ideal. Additionally, it would be nice if the lifetime of the fluorescent byproduct of $\beta$-gal activity was short, so I could detect a decrease in $\beta$-gal activity as well as an increase. Does such a fluorescent analog exist? I'm hoping to look at the switch from lactose to glucose utilization under the microscope, so I want lactose-using cells to light up and glucose-using cells to be dark, or vice versa. It does not have to be fully quantitative--a qualitative sense is fine.
I found this paper.
The work in the paper is cited in a more recent paper which reports beta-gal-based chemiluminescent imaging in mice.
It's unclear if any of this would be successful at a single bacterial cell level.
I posted this as an answer because it was too long for a comment. In general, your experimental approach is feasible, however, the devil is in the details.
Firstly, your biggest challenge will be to gain access to suitable optics. Imaging individual eukaryotic cells is relatively easy with any diffraction-limited optical microscope, because their scale is usually 10-20 $\mu$m. Bacteria are a much smaller cell size, and they (mostly) cannot be imaged using diffraction-limited techniques on a single cell level. To image a single cell requires super-resolution optics/techniques.
Secondly, it sounds like you're doing diauxic shift curves. It may not be necessary to image at a single cell level, and you may be able to get away with making good measurements of cells in aggregate using any camera-equipped microscope, or even an optical plate reader since you mentioned $\beta$-gal.
@Alan Boyd pointed out a chemiluminescent substrate for $\beta$-gal which could work well if you have access to an optical imager that can give you readouts of photon counts/second. This would give you readings of average enzymatic activity (which you can correlate to expression level).
Also, you describe having cells be bright/dark depending on the utilized substrate and that would be harder to achieve. There are many ways I envision to do this, but they all take some amount of genetic manipulation. Perhaps the easiest thing to do might be to make an expression plasmid that places LacZ under control of an IPTG-inducible promoter. These plasmids are very common and would give you an easy entry to image lactose usage.
Here is a link to on the Invitrogen/Life Technologies webpage detailing various probes to detect the activity of glycosidases, including $\beta$-galactosidase. These include DDAOG as mentioned by @Alan Boyd, as well as Fluorescein Digalactoside, Resorufin Galactoside, and Methylumbelliferyl Galactoside, among others. NB, this is an Invitrogen/Life Technologies website, so all the reagents they mention are ones sold by the company; there may be others not mentioned here. Of these, DDAOG is very stable, which would allow for measuring the initiation of $\beta$-gal activity, but might not work to measure decreases in $\beta$-gal activity (without bleaching the dye). Resorufin Galactoside is sensitive and stable at physiological pH, allowing measurement over a long period of time. Fluorescein Digalactoside is the most sensitive assay, but since it requires the removal of 2 galactose moieties for full fluorescence it's activation time may be slow.
For my particular usage (limited access to near-red frequencies), I think that Resorufin Galactoside or Fluorescein Digalactoside might be the best choices. However, given the cost of these probes, I might be better off just finding some lab with a strain with gfp under the lac promoter in the chromosome and asking them for a sample!