Many years ago, I visited the plant where Memorex was making all of their cassette tape. As one of the biggest suppliers, I was surprised to learn all of their tape was coated on a single coating machine.
The tape was made about a foot wide, then slit by a cylindrical set of knives. I don't know what material it was, but it was all one piece, and the engineer explaining it mentioned how they had to be replaced frequently and sent out for resharpening.
Water-jet cutting is the first thing that comes to mind. Laser might present problems with cutting speed, cost of equipment, and absorption properties of your material. (It's hard to cut a material that is transparent to the laser light.)
Ultrasonically driven knives have remarkable cutting properties and might be worth looking at.
Would it be possible to use a fine-tooth saw blade rather than a circular knife? Blades are available that are very thin.
Hot wire cutting is rarely a good solution, mostly useful for cutting foams rather than solids. In solids, the hot material can weld itself back together downstream of the wire, and it usually makes a ragged cut. Speed is likely to be a problem.
E-beam cutting requires doing the work in vacuum, which is highly undesirable.
Could the material be cryogenically cooled, scribed, and fractured?
Perhaps a guillotine system could be devised, where the cutting is between two edges, like a scissors (but rotating edges, of course, for continuous operation).
Perhaps the material could be perforated along the cuts, and then pulled apart.
How thick is the material that is being slit? Is it stacked or right off the roll or coater? You may want too look at other materials for the slitting blades. I've seen tungsten carbide circular knives that were used to slit coated polymer film. Other options could be Sialon (Silicon Aluminum Oxynitride) or ZrO2 knives. That material is used in the lumber industry. You may have to redesign the entire knife assembly to accommodate carbide or ceramic knives, but it should be easier to test than an entirely new technology.
You can also get coated blades. I would start with a google search on ceramic or carbide rotary knives or slitters. If cost per blade is not an issue, you can get knives coated with a variety of high wear materials.