You may find the most economic solution would be to buy a roll of wide photo paper from Ilford or Foma.
This would be by far the most practical and likely also cost-effective approach. The main limitation is the maximum roll width, although two (or more) bands of paper could be tacked/stapled/pinned side by side to a panel if a larger area needs to be covered.
Fortunately, all alt. printing processes work fine with UV-A. UV-B is the band that starts at something like 350nm, downward. You really don't want to cast any significant amounts of UV-B onto players or an audience - especially the players would receive levels of exposure that with UV-B would be really problematic, especially for their eyes (look up 'Klieg eyes'). Having them wear sunglasses would help.
Anyway, all UV-sensitive alt. printing processes are really far too slow for this application, so that's a dead-end street to begin with.
As to the idea of coating a more common silver halide emulsion, the following scenarios come to mind:
1: Pre-coat several sheets of cloth or whatever it is that the stage design requires in a batch-wise fashion. Then use one sheet for each performance. Coating, storage and mounting of the coated sheets would have to be done under safelight conditions. The bath-wise coating of sheets would likely be a little more efficient and thus less costly than doing it one at a time for each performance (it's also less disruptive to the work schedule, I imagine).
2: Modify the existing gelatin-based emulsion to a spray coat application approach. There's two ways I can potentially see this working, at least at a theoretical level:
2a: Spray coat a warm emulsion so it remains liquid. The most practical (insofar we can call it that) way of doing this would be to work in a heated cabin (e.g. 45C) so that everything the emulsion touches remains above the melting point of the gelatin. A challenge would be to bring the coated sheet outside the coating chamber for it to cool down, set and dry. Another challenge is that even at elevated temperatures that are already very uncomfortable to work in, the rheology of the emulsion will likely present severe problems for spray application.
2b: Dilute an existing gelatin-based emulsion in a molten state with demineralized water, to a level where the cooled-down emulsion remains liquid and sprayable. Gelatin solutions tend to be liquid at low percentages of e.g. 0.2% - I imagine a dilution of
at least 1+50 to 1+100 of the commercial product would be necessary to obtain a sprayable emulsion. Application would then be done by spraying many consecutive layers of emulsion, drying in-between, until the desired density is reached. Evidently, the repeated spray coating of the same sheet would be very time consuming.
3: Design an entirely new emulsion from the ground up, replacing the gelatin with another colloid. Coincidentally, someone has been inquiring into this on this forum
https://www.photrio.com/forum/threads/can-anyone-help-me-make-emulsion-without-gelatine.201485 The gist of that thread, however, is that there are lots of ideas, lots of challenges, and it's all still firmly in the paper-napkin phase, not to mention that spray coating is not a requirement in that scenario, which would pose an additional set of profound challenges.
One thing that comes to mind is that the requirement of spray coating is pretty much antithetic to the nature of a silver halide emulsion. Such an emulsion in practice is really more of a suspension, and concentrated suspensions tend to spray very badly due to their high viscosity. Performing the R&D to solve this fundamental issue sounds quite unrealistic as a work package within a theatrical production. It's the kind of thing you'd expect a firm like FUJIFILM to dedicate money on - and they certainly would have, if it had made any economic sense (which it apparently doesn't!)
So that's a long way of saying: give
@Don_ih's idea above a thought. It's the best shot you've got.