No, mahogany grain filler won't make the pieces of wood match. All the wood grain is in the same direction, but there's that one piece which has an additional very noticeable cross-grain pattern, very much like the flame pattern in flame maple.
If your friend insists on a stained finish with clear over the top, then they'll have to live with the mixed wood patterns.
To get something similar, I'd probably try out a translucent finish, with an amber clear coat over a thin pale yellow or cream base coat. But you'll probably need to use nitro sprays to get the tinted lacquer. I did this last year when refurbishing a telecaster body to get a "TV yellow" style Fender butterscotch finish. I probably would go a bit thinner on the base coat if I did it again.
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