Quite honestly the best places to hear fado here in Lisboa do not really come with stellar cuisine. While even some of the more formal places tend to have better food, you won't necessarily get very innovative cuisine.
Probably the most chic one is Sr. Vinho in Lapa/Madragoa
There are also more "typical" Fado Clubs/Restaurants mainly in Alfama and the Bairro Alto
Sr. Fado - Alfama - you can make a reservation online
Adega do Ribatejo - Bairro Alto - a bit touristy
Clube de Fado - Alfama - Well-known and very touristy
Tasca do Chico - Bairro Alto - an informal bar to hear fado
Tasca do Jaime - Graça - Simple, homestyle cuisine and genuine Fado for non-tourists
Grupo Excursionista ‘Vai Tu’ - Bica - Informal Place to hear fado as well - Not touristy and more of a club/association
The Museu do Fado has a link with other restaurants/clubs/associations that might be helpful.
Please let me know if I can help in any other way.