(Face fabric is black, cotton twill)
I don't mean to be uncharitable, but I've never seen nor heard of a black cotton frock coat. All of the black coats I've seen were made of fine wool. Summer frocks existed, but it's my understanding they were generally made in light colors to repel heat, and linen was more common than cotton. Is it too late to revisit your choice of fabrics or has the coat been cut out?
If you are committed to cotton, you might want to consider making the coat as a "summer frock" even though it's a dark color. In this case no lining would be required, but the seams should be felled to prevent ravelling. You can starch the coat to make it retain its shape without lining or padding, and it will be cooler to wear.
Hope this is helpful,