When one begins the process to purchase an M1 Garand from the CMP, you have to meet certain eligibility requirements:

Proof of U.S. Citizenship
Proof of Age
Membership in CMP Affiliated Organization
Marksmanship or other Firearms Related Activity

On the last point, you can use a C&R FFL (from the CMP website: "Firearms Owner Identification Cards that included live fire training. - FFL or C&R license.").

If you purchase an M1 Garand and use a C&R FFL to meet the last point, are you effectively using the C&R FFL to purchase the M1 Garand, thus requiring that you log it into the C&R FFL bound book?

The rules are the same whether you buy from the CMP or anywhere else. You ARE licensed, and a requirement of that license is that you have to log EVERY C&R gun you buy, regardless of whether you showed the seller your license.

If it's 50 years old or on the C&R list, it gets logged. Period.

Log it.

Yes, you're supposed to log it. Besides, it's just a good record to have. If your rifles are lost or stolen, you have a record of the serial and date of purchase for insurance purposes (I speak from person experience )

The biggiest misconception of the ffl 03 is "you use your license ". There is no such thing as using your license.  If you purchase a firearm that is C&R eligible then you log it, period!  Even if you go to Big 5 and buy a Mosin nagant you are still required to log it!  I own two CMP garands and both are in my book