As you pointed out, Pilgrim's Progress was and is a very anti-Catholic book. Seeing as its author, John Bunyan, was a Baptist writer serving time in jail for opposing an Anglican church in England at the same time England was living in fear of the even-more hierarchical Roman church, you can understand why he might be anti-Pope.
The basic tenant of the book - that the Christian life is a journey that must be accomplished on one's own, is as about as Baptist / Puritanical as it gets. Like all Puritans, he celebrated rather than suppressed emotion, and wanted to ensure the "personal relationship" with Jesus took center stage. (Yes, Puritans were not the dour people popularly portrayed. Want to read a theology that celebrates sex between man and wife? Look to the Puritans - not the Catholics!)
These two biggies - the unmediated experience and the anti-hierarchical church polity - are probably the biggest bugaboos for a Catholic.
Beyond that, the book presents salvation as an act of free will, and seems to completely discount election or predestination. These issues may also cause some interesting discussions.
That said, this is the second best selling book of all time, just behind the Bible. Even if a good Christian were to disagree, I'd suggest it is edifying to read and think. The praise of Acts 17:11 comes to mind:
Now the Berean Jews were of more noble character than those in Thessalonica, for they received the message with great eagerness and examined the Scriptures every day to see if what Paul said was true.
As a Baptist believer in soul competency, Bunyan himself probably wouldn't have wanted you to accept his opinions without considering them against scripture. His genius was to rehabilitate allegory from static symbolism and push the metaphor into dynamic journey and learning that could be used to direct a believer's life.