The Media Type guides the platen gap, the head adjustment (viz standard, wide, etc.) and the ink laydown; it also affects profile behaviour in the sense that the Media Type you print with should be the same as the Media Type specified for the profile, but in this case, understood you have none. Cotton Artist Textured (ICAT) has no OBA content and it's a thick, heavy paper, while Enhanced Matte is loaded with OBA and is a much thinner, lighter product. USFAP has no OBA, a tint closer to that of ICAT and a weight and thickness closer to those of ICAT. So, all told, you would probably be better off starting with USFAP as the Media Type in your 3800 and see what it produces. But if you are using application colour management (LR or PS manages colour) you still need to load a profile in the printer driver. You could start with the P800 profile for ICAT and see how it performs. It would differ from a 3800 profile for the same paper to the extent of the main difference in the inksets between the two printers.
I think you are in a situation where if you want to use papers that no longer cohere with your printer because the printer is no longer being supported, and this fix of using the P800 profiles doesn't work well, you will need to either get custom profiles made for the papers you will be using, or upgrade your printer.