"Dollheart" is the English translation of the French "Cœur de poupée". This was indeed one of the nicknames of King John, and the poupée in this context does refer to the children's toy or puppet.
As with most nicknames, we will probably never know who coined it, but according to Jacques Choffel, in the epilogue to his biography of John's older brother King Richard I, Richard Coeur de Lion it was used by his contemporaries to compare him unfavourably with his brother:
"Cœur de Poupée, diront les barons poitevins, par dérision, en souvenir de son glorieux frère auquel amis comme ennemis reconnaissaient un cœur de lion."
In derision, the Poitevin barons named him "Doll's heart", in remembrance of his glorious brother, who both friends and enemies recognised as a lionheart.
As for 13th century sources, the nickname is certainly mentioned in the Chronique rimée by Philippe Mouskes. He cites his source as follows:
"le roi Johan, que G Guiart appelle cuer de poupee"
Which would seem to be a reference to chronique métrique de Guillaume Guiart. Writing about the year 1214 he says:
Va s'en Jouhan Cuer-de-poupée
Ne pense à honte n'à laidenge;
Sa gent après lui se défrenge;
Tout fussent-il, pour eus combatre,
Contre un des François plus de quatre,
N'ont il talent qu'il les atendent.
but sadly didn't cite his sources.