PCB materials have improved greatly over the 36 years I have been doing designs (especially Z axis CTE). Now you can get what you want, as Klaus says, the first stop is your PCB manufacturer. Expect to pay a bit more when you go outside the norms...
FR4 is a generic term these days, have a browse round ISOLA and other laminate manufacturers for the available choice of materials, there are many, not just variants of FR4 , but more exotic (and expensive) solutions.