Assume one constructed a submersible such as what Captain Nemo commanded, the Nautilus, and it had a useful depth of 1km. Using 19th century technology, what would they need to develop to allow functional and relatively practical exterior lighting? They intend to observe and explore the seabed near volcanic vents.
Verne's divers (before their undocumented demise from caisson disease) used battery powered Ruhmkorff coils (auto-induction high voltage suppliers, similar to modern ignition coils) to provide high voltage for Crooke's tubes as portable lights (presumably with very heavy tube walls, perhaps of fused quartz). There's no reason these couldn't be made larger, powered by the boat's internal electrical system, and mounted in steerable reflectors for exterior lighting.
Mercury and sodium vapor lamps used in outdoor lighting until the 21st century (when LED replaced them) operate on the same principle, and have the same problem: the small number of bright emission lines makes color perception very inaccurate. With mercury arc, you can at least use a phosphor screen (excited by the UV) to provide a broader color spectrum (and this was known in the 19th century as well)
The fundamental problems normal light bulbs would experience under high pressures combined with an incandescent or combustible light source are related to compressive and tensile forces acting on the casing and glass envelope. The casing can be made of any suitable metallic alloy available at the time, such as the brass used in diving helmets. But the glass will be subjected to unusual stresses. So the manufacturing process of that glass would have to evolve in a way that produces amorphous, crystal-free glass.
Arc furnace foundry
Getting the material to melt in a homogenous state, avoiding any crystalline inclusions, has been achieved using a furnace that employs an electric arc for its heat source. They would have to develop this technique, and also experiment with various metallic glasses. That had already been an art for the purpose of making rarefied glasses such as the Crookes tube that led to the discovery of X-rays.
High pressure quenching
The inventor needs to not only form the homogenous glass material amorphously, but they need to quench the glass at high pressure, which will further align the glass microstructure amorphously.
It is likely that you will only produce small lenses with this method, so the final lamp may be comprised of several compound windows in a convex honeycomb frame.