Copy Control (which is the only on-disc copy protection for music CDs that I know existed) should be no problem at all. I use EAC (Exact Audio Copy) to rip my music CDs, and there you don't even notice whether a disc is protected or not. Everything works just as well as if it isn't. I would have supposed that MediaMonkey would be able to as well. Try ripping with EAC and see if you can get around the problems.
Since you didn't clarify what problems you get, it's not unlikely that it's not the copy protection itself that is giving you trouble, but its side-effects. These schemes always make use of scrambled error correction data, which means that copy-protected CDs are more susceptible to scratches, dust, wear, etc. While a real Audio CD might have enough information to correct read errors, a respective Copy Control disc is always of lower quality, so depending on your drive and the condition of the disc it might be enough to make it unreadable.
I never even had problems with copy-protected CDs when they were still being made. The only indication I have that they ever actually did anything was that I heard about other people who were outraged, leading to audio CD copy protections getting the mercy of being killed off after having been used for just about a year. (Wikipedia says 2001-2006, but I have never seen a Copy Control disc outside the range of 2003-2004, so they must be rare).