Why did Lovett invite Rose aboard his search boat in Titanic?

Lovett, obsessed with the Titanic and the "Heart of the Ocean", has been searching for the diamond for three years and the time and money for his expedition were running out, so he clearly is in a difficult position. His investors want to call off the search if he does not come forward with any progress soon.

When they find the picture showing a woman wearing the diamond the day the Titanic sunk, he realizes that after years of searching this is the greatest shot he ever had on finding out the location of the diamond.

So in the light of this great discovery, the cost of flying Rose in and interviewing her personally is nothing compared to the expenses this expedition probably devoured in the previous months tapping in the dark.

As you said, it is likely that Rose bargained her way into the flight, offering information/her story in exchange for seeing the Titanic and especially her picture again. It is unlikely that she had a hard time convincing Lovett though, because he wouldn't want to risk missing important details by disgruntling his only contemporary witness and he probably was eager himself to hear the story from face to face.

So the main factor for this deal was the fact that Lovett was running out of time.

Of course it seems a little bit unfair from Rose's side to lie about the diamond, knowing exactly that this is what Lovett and his crew are searching for for years. She carries it with her the whole time, and finally drops it into the ocean to find closure for herself while she could help a man struggeling for his "career".

On the other hand with her story she changed Lovett's view on the Titanic and her history. He was so obsessed with the diamond that he never thought about the real meaning of the events and the disastrous impact it had on so many lives:

Three years, I've thought of nothing except Titanic; but I never got it... I never let it in.

So it might be argued that while she did not help him find the diamond, she helped him find something better: the true story behind the Titanic.