I checked my working Beckwith genealogy and agree with your line as John Chipman-John-John-James-Matthew-Matthew. I have nothing on John Chipman Beckwith other than a birth date of 27 Dec 1865.
I will add your information to my listing from John Chipman Beckwith to the present.
Sorry to be of no more help than this.
By the way- Matthew of CT (the immigrant) had a wife named Mary (not Elizabeth as shown in most genealogies). His birth year is c1612 (no evidence to indicate a borth for Matthew of CT of 1612 in Ponterfract (or however it is spelled) as son of Thomas or Marmaduke. If you do have any evidence for his actual birth date or names of parents, I would appreciate your sharing it.