Very good start Neil! I wondered what it would sound like in first person. (I thought the second line might make a good first)

Barflies raised their glasses shouted cheers
When I came to the bar in search of beers
Billy asked how I was keepin'
I said not well my wife is leavin

I'm not crying in my beer
I just turn the other cheek
I'm not crying in my beer
Tears make it too damn weak
I'm not crying in my beer
You can't fix a heart with grief

Optional last lines:
(Though I know how it may seem)
(Is that so hard to believe?)
(Not my style-that ain't me)

Or as your observer:

Barflies raised their glasses shouted cheers
When Bill came to the bar in search of beers
I asked Billy how the hell ya keepin'
He said not well cause my wife is leavin

He's not crying in his beer
He just turns the other cheek
He's not crying in his beer
Tears make it too damn weak
He's not crying in his beer
You can't fix a heart with grief

Optional last lines:
(Is that so hard to believe?)
(Though I know how it may seem)
(Sometimes looks...they can deceive)

Best of luck!
-Mike