If you are asking if there are lineages to Charlemagne that do not go through the Magna Carta Barons, the answer would be: of course there are.
I'm putting this out here to help those who may have already discovered their link to one of the following eight MC barons, by showing where the documentation is that will take them from their Magna Carta baron to Charlemagne. I'm sure there are many paths to Charlemagne that do not go through England at all, but what we have in the books I cite by Douglas Richardson, are paths with documented evidence.
Michael, when I discovered Magna Carta barons in my family tree I made it a priority to document those connections. I was amazed to eventually discover that many of the Magna Carta Surety Barons are descendants of King William the Conqueror, and so also descended from Charlemagne. I would have thought the same as your comment if not for the years of considering those lineages - and also for the gifts of nine not-so-inexpensive books. I have not found connections to Charlemagne for all of the MC barons, and the following is not a complete list.
My birthday is in December, so that year I asked to have no other presents for my birthday and Christmas than the four volume set of Magna Carta Ancestry, by Douglas Richardson, published 2011.
Richardson published Royal Ancestry: A Study in Colonial and Medieval Families, in 2013. It begins with William the Conqueror (1066) - and includes the Magna Carta barons. The Five volume set was my birthday/Christmas gift a few years ago.
In the Introduction to Royal Ancestry, Richardson writes that
"The Appendix in Volume V sets forth descents for both King William the Conqueror and wife, Maud of Flanders, from the Emperor Charlemagne (died 814).
How many of the Magna Carta barons were descended from William the Conqueror, and therefore from Charlemagne?
One random peek (see page 557) into Volume One of Royal Ancestry shows that Walter FitzRobert of Little Dunmow, Essex, who is father of Robert FitzWalter, Magna Charta Baron, is descended from Alice of Normandy, wife of Lambert, Count of Lens.
Richardson includes Alice (aka Adeliza) of Normandy as a full sister of King William the Conqueror, who is a descendant of Charlemagne. Following is an incomplete list of Magna Carta barons with documented lineage to Charlemagne.
Hugh Bigod and Roger Bigod are also descendants of Charlemagne through Alice of Normandy, Royal Ancestry Vol 1, page 362.
Richard de Clare (died 1217), married Amice of Gloucester - she is a descendant of King William the Conqueror. Their son Gilbert de Clare (died 1230) is therefore a descendant of Charlemagne. Royal Ancestry Vol 1, page 216.
Saher de Quincy is a descendant of Alice of Normandy, Royal Ancestry Vol !V, page 436.
If you are a descendant of MC Surety Baron Geoffrey de Say (died 1230), then you are a descendant of Charlemagne through Geoffrey's wife Hawise de Clare, whose mother, Amice of Gloucester is a descendant of Charlemagne. Royal Ancestry Vol IV, page 562.
MC baron William D'Aubengy (died 1236) is descended from Charlemagne through Alice of Normandy, sister of William the Conqueror. Royal Ancestry Vol II, page 392-393.
Sir William de Huntingfield (died about 1221), Magna Carta Surety Baron, is a descendant of Charlemagne through Alice of Normandy. Royal Ancestry Vol III page 373-374.
Mr. Marlyn Lewis, creator Our Royal, Titled, Noble, and Commoner Ancestors and Cousins, has a convenient list of the Magna Carta Surety Barons, linked to their profiles with footnotes to Royal Ancestry and Magna Carta Ancestry. by Richardson.