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I sometimes carry an FEG PA-63, which is a Walther PPK clone and almost identical in weight, size, and shape and operation to the Bersa Thunder .380. It's chambered in 9mm Makarov, a slightly longer, wider, and hotter round.

In answer to your question, I carry with a round chambered and a full magazine behind it and the safety on.

As with the Thunder, it is DA/SA. I'd have no qualms about carrying mine hammer-down and safety off as someone else mentioned above - there's no way that DA trigger can be accidentally pulled. The stock DA pull on the PA-63 is said to be in the range of 19-25 pounds, and I don't believe that to be an exaggeration. About half of the people I take shooting for the first time are unable to shoot it in DA because they lack the strength to pull the trigger completely. I've trained with mine drawing from safety on, so there's no reason for me to change the way I carry. From what I've read the Thunder has a DA pull of about 7 pounds (though pretty long), so the situation may be different for you. I also don't know how much resistance the safety is, but I believe the general consensus is that having a round chambered, hammer down, and safety on is the norm. Lots of practice drawing should make disengaging the safety a non-issue.

Carrying without a round in the chamber makes your gun useless in so many situations where a quick draw and fire or single-handed operation is required. You're also reducing the amount of ammo you are carrying by one in an already limited gun.

If you are concerned about carrying with a round chambered, I would recommend some range time to get more familiar with the operation of your gun. Remember, it's not going to "go off". Modern firearms don't do that. The only reason it's going to fire is if you pull the trigger.