The comment by K. T. gives me a nice idea.
Set up your reaction in an environment with stable illumination and against a white background or far from any specific one. Then you can take ordered photographs that you print on transparent slides.
You can also set a cuvette in a white box and fill it with sample from the reaction vessel, to take better photographs.
After you cut them in stripes with the cuvette face size and by inserting them in the slits of the spectrophotometer you get the absorbance spectrum of a sample that mimics your reaction.
You should easily get stripes of the cuvette size, if not you can always arrange something to be measured against the appropriate background (baseline).
You can also make standards or at least checkpoints using a single component of the reaction, and in all cases measuring at a relevant wavelength.
The only hypothesis is that the A of the stripes scale linearly with that of the reaction solution, and this can be confirmed as in the above paragraph.
I never thought before of this especially because my work required some accuracy. But I could have tried it indeed when the kinetic study apparatus was already in use (just to identify a change in colour, for instance). For your essay, it might be more than enough.
If a student of mine would circumvent the given limitations in experimental set-ups using a trick as I have described, I'd be rather impressed.
edit I must add that once, for fun, I used the smartphone to photograph my finger at different levels of blood oxigenations, and after emphasising the differences looked for a trend, which I have found. This made me consider possible colour picker applications or photography suites as a tool to quantify the color (absorbance) without even going to a spectrophotometer. Digging Internet with this in mind I have found that my idea isn't original anymore and found project similar to that in the Q, and indeed conducted using a smartphone. Examples can be easily found and downloaded searching for "smartphone as a colorimeter" and similar search keys.
https://pubs.acs.org/doi/abs/10.1021/ed300567p?src=recsys&journalCode=jceda8
is an introduction in J. Chem. Ed.