I believe one of the reasons you might have difficulty researching this family is that the family name was changed from Hardtman to Berkeley prior to John's birth.
John HH Berkeley's father Thomas Berkeley Hardtman changed his surname from Hardtman to Berkeley, which was the maiden name of his mother Ann Howard Berkeley (daughter of Maurice Berkeley) after her death. This made his new name Thomas Berkeley Hardtman Berkeley. So his children's surname became Berkeley instead of Hardtman. Thomas Berkeley Hardtman Berkeley's siblings, and their descendants, continued to carry the surname of Hardtman.
Below is an abstract of his will, and along with the names of his children, including John.
3. Thomas Berkeley Hardtman-Berkeley (son of Thomas Allman Hardtman) was born on 14 Jan 1824 in St. Kitts and died on 6 Nov 1881 in Cedar Hill, Antigua, at age 57.
General Notes: Born at the Farm Estate, Sandy Point ======= Will of Thomas Berkeley Hardtman Berkeley 1879
Honorable Thomas Berkeley Hardtman Berkeley of Millikon, St. Christopher, CMG, planter, formerly known as Thomas Berkeley Hardtman. Will dated April 7, 1880. "My sons, John Hart H. B. of St. Christopher, planter; Honorable Henry S. H. B., activing Attorney General of the Leeward Islands, and Humphrey Goerge H. B. to be Executors.
To my two sons, John Henry and C. Arthur Shand, Esquire, on trust, my meiety of Ottley's (or Oltley) estate in Trinity, Palmetto Point, my moiety of Greenland estate in St. Mary Cayon, and land and buildings in Pall Mall Square, Basseterre, now used by the Estates Agency Depot, and all stores to sell and to form part of my person estate. Accounts with James Ewing and Co. My daughter Alice Howard, wife of Charles A. Shand, of Fitches Creek, Antiqua, planter, 2000 pounts. My daughter, Annie Berkeley, wife of the Honorable A. W. Holmes A. Court, 2000 pounds. My daughter, Aymee Martha, 2000 pounds at 21. To my son, John Hart, Shadwell estate, charged with Annuity of 100 pounds to his mother. To my son, Henry Spencer, Cedar Hill estate, subject to like annuity. To my son Humphrey at 26, Foundtain estate. To my wife, Alice Hart, 100 pounds. My sister, Ann Isabella, 20 pounds a year. All resideu to my three sons, John, Henry and Humphrey. Witnessed by W. E. S. Napier, John B. Blizard. Will was proved on November 15, 1881.
Death: At Cedar Hill, Antigua, on the 6th instant, the Honorable Thomas Berkeley Hardtman Berkeley, C. M. G., President of the Federal Council of the Leeward Islands, age 57 yeras. From a long notice which follows it appears that he died in a few days of fever. He was a very successfull planter and attorney of many estates, and for over 30 years a M. of the local legislature of St. Kitts. He was also Vice President of the Federal County, and a few days before his death was nominated President on the retirement of Sir Oliver Nugent. The C. M. G. was conferred in 1879. (The St. Christopher Gazette and Charibbean Courier (sic) ).
Thomas married Alice Hart Rawlins on 26 May 1846 in Basse Terre, St. Kitts.
Children from this marriage were:
+ 13 M i. Thomas Berkeley Hardtman-Berkeley was born on 20 May 1847 in Probably St. Kitts and died on 7 Dec 1876, at age 29.
+ 14 M ii. John Hart Hardtman-Berkeley was born on 1 Aug 1849 and died on 12 Aug 1910 in Brighton, at age 61.
+ 15 M iii. Henry Spencer Hardtman-Berkeley was born on 6 Sep 1851 and died on 30 Sep 1918 in Bures, Suffolk, England, at age 67.
+ 16 F iv. Alice Howard Hardtman-Berkeley was born on 9 Dec 1853 and died on 14 May 1910 in Antigua, at age 56.
+ 17 F v. Anne Berkeley Hardtman-Berkeley was born on 31 Jul 1855.
+ 18 M vi. Humphrey George Hardtman-Berkeley was born on 13 Oct 1860 in Millekins, St. Kitts, died on 12 Jun 1913 in Suva, Fiji, at age 52, and was buried in Buried At Sea.
+ 19 F vii. Aymee Martha Hardtman-Berkeley was born on 31 Jan 1868.