Were Catholics originally offended by the Sacred Heart? According to a comment by Mr. Hayword, the Sacred Heart was something which was originally opposed in Catholicism, but I have not found clear evidence of this, only a citation of an Anglican who cites Orthodox priests whose criticism are not undisputed (you might look at the comments here). 

This does not provide analysis of WESTERN condemnation of the practice, though, or even Western censure. Does any such document exist?
By the 1796 Janenism, which disdained devotion to the Sacred Heart, was very roundly condemned by Pope Pius VI in Auctorem Fidei one paragraph stated:


  
  The doctrine which rejects devotion to the most Sacred Heart of Jesus among the devotions which it notes as new, erroneous, or at least, dangerous; if the understanding of this devotion is of such a sort as has been approved by the Apostolic See,—false, rash, dangerous, offensive to pious ears, injurious to the Apostolic See.
  


So, prior to this, in the nearly 8 centuries of practicing the devotion in the Church, there may have been some who said made a big todo about those who over-did it.  Jansenists, who started as a movement within the Catholic Church, were certainly among them.  They'd say that it's inappropriate to worship a portion of Jesus, especially a wholly material and fleshy part. 

But seriously, in reading through the Catholic Encyclopedia article and not finding one condemnation of suspicion of the authentic devotion.  I doubt that the cult was ever suppressed (if that's the right thing to say about Our Lord) like it might be for some saint of dubious origin or some aesthetic practice taken too far.