Just be aware that DNA tests are not 100% accurate, and in some cases can be way off. Some also don’t have the staffie breed. So although it may help with convincing authorities (I’m not sure if it would hold up in court; that might be something you should look into before spending the money) that it isn’t an American Pit Bull Terrier (which I don’t think any tests at the moment have), it probably won’t give you a good indication of the actual breed.
Without papers, there is no 100% way to tell what breed your dog is.