Is a 100 amp transfer switch adequate for a 15KW generator?
Installing a 200A transfer switch will do you no good unless you're planning on buying a 48KW generator soon. Your transfer switch is suitable.  Your generator cannot produce 100A so you need to be watchful of what loads you switch on, careful that neither phase exceeds 62.5A and that the two phases are in balance within the limits in your generator documentation.   If you don't do that your generator can brown out or stall.
If you use the 100A transfer switch with 100A breakers you need to use a feed wire suitable for 100A between the generator, switch, and panel.   You could use the 100A transfer switch with 60A breakers, then you can use lighter feed wiring and you'll pop the breakers before browning out.  Even then you have to watch the balance of the phases.