Both versions are gramatically correct and mean the exact same thing.

Both sentences have no issue with the meaning, here is about the subtle difference:

• la = complément d’objet direct (COD)
• à sa propriétaire = complément d’objet indirect (COI)
• en un morceau = complément d’objet second (COS)

In general we are used to place in order COD-COI-COS. Just because it is a natural order:

(object of the discussion)- (to who/what it is directed)- (a superfluous comment)

But this is not a rule and you are free to emphasize the part you wish.

• Je te prie de la rendre à sa propriétaire en un morceau is what we would naturally say
• Je te prie de la rendre en un morceau à sa propriétaire this will lead to think that the part en un morceau is, for the conversation, particularly important.

I would add to the other answer thats i’ve never heard anyone use “en un morceau” the correct expression is “en un seul morceau”. It has the same meaning and people will probably understand you. But people, like me, will be surprised by the lack of “seul”.