Multi Genre Cover Band (Going Original)

So you can see from the title of this thread that I'm writing about a band that I'm in.

https://www.facebook.com/pages/The-Dinner-Jackets/129793117095074

Our Band Includes 6 Members

Sax Player/ Singer
Two Electric Guitar Players/ Singers
Acoustic Guitar Player/ Percussion/ Female Vocals
Bass Player
and
Drummer/Singer

We try to play anything. No matter it be what the audience wants us to play, what we want to play, or what clients want us to play when we are hired. So we play a wide variety of songs ranging from the genres of jazz to punk, or blues to hair metal, or funk to country.

This seems to go over very well with audiences we play for due to the fact we choose our set based on the vibe of the crowd or the other band(s) we are playing with.

But personally we are finding this rather difficult for writing our own music. We have different genres for each song on our upcoming EP we will be recording this summer. Some are reggae, some are folk, some are alternative, ect...

So I come to this forum for advice, is it a good idea having a set of original music that varies in genre?

Some bands are all over the place and can get away with it. There has to be something at the core of each song that makes it "you." Think of all the stuff The Beatles did. From Chuck Berry to Motown to where ever. But you always knew you were listening to The Beatles. If you establish that core identity, then you can do anything with it.

I say go for it, mix it up, not many bands can do that well and if you can its gonna be a point of difference between you and most other bands that just stick to one genre. Play to your strengths man. Personally I would much rather go see a band that can play a few different styles.