Are the dancers you perform for interested in Strict Tempo and ballroom dancers or are they just "social dancers"? Our local art center sponsors both ballroom classes and dances. Recently they had a big band play, but as most of us were ballroom dancers there was a disconnect from what they thought was tango, foxtrot, rumba etc. We've also had some live groups have a hard time keeping the tempo even for the whole song. That can be frustrating and challenging.

I think your question is wonderful and thoughtful. If your audience is people who like social dancing, they'll be much less particular. I agree, listen to almost anything by Sinatra, Michael Buble or Tony Bennett. You'll get lots of foxtrots from them, but people enjoy those. My favs are "Come Fly With Me", "Summer Wind" and "Just the Way You Look Tonight". For a rumba, "Girl From Ipanema" is good. Waltzes are also popular such as "Moon River", "Try to Remember" or "Sunrise, Sunset". There are also plenty of modern songs that fit these categories, but you said tea dance so I'm thinking the group is of a certain age.

Unfortunately most of the live bands trying to play strict Tempo haven't a clue what is the difference between a Modern Waltz and a Viennese Waltz just to mention two. Modern Waltz 28 BPM Viennese 60 BPM. The correct Tempos for all the strict Dances can be found by Google the Dance Sport Site in whatever country you are living. Try Dance Sport U.K. or Australia. Find your way to rules and then Tempos. I find that with live bands they have Vocalists and play to suit the Singer and not the Dancers. That is why we get a fluctuation in the Tempo being played. They don't seem to understand that the beats being played must be constant from the Four Bar introduction right through to the end of the tune.

This book (PDF): http://www.mrk-inc.com/users/fjacoby/ballrm0.pdf was written by a band leader who also dances. It's a "how to" guide for other musicians on playing music for dancing, and it devotes considerable space to discussing why the correct tempo matters to dancers.