I see Frederick left a will when he died, a rich man, in 1906.
Now that would be worth saving up and sending for, to see if he makes any mention of his second family. If so, you have your proof easily.
Probate index entry-
Frederick Banks of the City Asylum Lodge Road Birmingham died 3:Nov 1906 Probate to Charles Banks Gentleman 
Effects £19,916  10s  8d