Actually good style is to say things in a natural way and not in the most complicated way. So I would formulate:
He went to work though he was sick
Being sick, he, nevertheless, left home to go to work.
Such a sentence has the smell of school exercises, i.e. sentences are generated without conveying an idea and feeling what is natural and simple language.
In your sentence I would see the following flaws:
1 Transforming a simple subordinate clause into a participle construction is a stylistic figure of elevated written language. Have you heard participle constructions in colloquial language?
2 The two contrary ideas - sick and going to work - clearly express the idea of oppposition. So "nevertheless" actually isn't necessary.
3 If you go to work you normally have to leave your home. So to state "he left home" to go to work is a banality.
Simple sentences would be:
1 He was sick but he went to work (all the same).
2 He was sick, nevertheless he went to work.
3 Though he was sick he went to work.