If you put the other last name in the "current last name" field, that profile would appear as James (Bloodsworth) Bloodworth, and turn up in searches for either name. I'd stay away from doing it this way unless the person in question was born a Bloodsworth but became a Bloodworth over time. A clerical error isn't the same as a name that was commonly used by the ancestor.
Since WikiTree links family members to each other quite clearly, a compromise may be to use your discretion regarding available records and decide which profiles will have Bloodsworth as the last name at birth, and whose children were Bloodworth. This way, anyone searching for either name can easily click through to see that Daughter Bloodworth's father was a Bloodsworth.