Acadian Genealogy is so prevalent that the best way to find these people is just to search for them using quotation marks. For example, for Anselme Bastarache, search for:
Out of the may web sites that discuss his family to varying degrees, there is:
outlining his ancestry. There are also:
an so on. Granted these are hardly as worthwhile as actual research, but at least you should be able to get some guidance.
I do not see anything and wonder if you do not seek Dosithee Gallant.
I can place Gilbert Bastarache and his wife, Catherine Gallant, in Shediac Parish, Westmorland Co., NB in 1906 and 1909. By 1911 they seem to be in Dalhousie, Restigouche Co., NB. In 1916 they are in Five Fingers, Restigouche Co. In Dalhousie Parish, in the 1911 census, Gilbert Bastarache was b. MAY1877 and Catherine, his wife, was b. SEP1880. Children were:
Alice b. SEP1901
'Livail' (son) b. NOV1902
'Eulia' (daughter) b. NOV1903
Yvonne b. DEC1906 (Mary Bastarache b. Shediac Parish, Westmorland Co., NB 25DEC1906)
Ellen b. JAN1909 (Mary Alice Bastarache b. Shediac Parish, Westmorland Co., NB 2JAN1909)
Deliema b. APR1911 (Rose DeLima Basatrache b. Dalhousie Parish, Restigouche Co., NB 2APR1911)
All were b. New Brunswick. A later daughter was M. Agna Irene Bastarache b. Five Fingers, Restigouche Co., NB 1FEB1916.
Gilbert Bastarache, b. MAY1877, may have been Gilbert Bastarache b. 2MAR1877 in Wellington Parish, Kent Co., NB in 1901, s/o Arcene and Clairisse Bastarache. The variations in spelling the surname make it difficult to do an effective search.
Trying another approach, I tried looking for Catherine's brothers, Alfred, Andrew (Andre?) and Godfrey Gallant in the 1901 census of NB. The only Godfrey I found of interest was in Shediac Parish, Godfrey Gallant (26NOV1883), s/o Martial (12AUG1852) and Christienne (24APR1848) Gallant. In Kent Co., NB in 1912, Godfrey (Godfroi) m. Suzanne Bourque. His marriage record names his parents as Martial and Christine (Gallant) Gallant. Martial (b. Grande Digue, NB 12JUL1851, d. Grande Dique, Kent Co., NB 21DEC1933) and was the s/o Milieu and Henriette (Arseneau) Gallant.
Of the Alfred Gallant's one interesting was Alfred Gallant b. Grande Digue 11FEB1869, d. Kent Co. 13JAN1949, s/o Dosithee and Marie Blanche (Caissie) Gallant. He also had a brother Albert Gallant. His wife was Marie Babineau.
This appears to be the correct family. In 1891, in Shediac Parish, Westmorland Co., NB, was the family of 'Dosite Galleng' (b. ca. 1842.) The spelling is atrocious and the image faded. At best I make his wife to be Marie Blanch (47) and children to be Alphee (22), Andre (16), Jean Q. (son, 15), Elizabeth (13), Chatherine (10), Joseph E. (8), Joachim (6) Alber (Albert, Alfred's brother, 4) and Vital (2.)
In 1881, in Shediac Parish, he is indeed Dusty Gallant (42) but I can only attribute that spelling to an English enumerator who had no clue. Especially as nearby lived the family of Dusty Poirier. To find two Acadians within a few households of each other both named Dusty is remarkable. To find two Acadians that close named Dosithee is unremarkable. Presumably the spellings of the forenames of the other members of the household have been anglicized as well. These were his wife, Mary (38), and their children: Alfred (12), Andrew? (7), Godfrey (5), Elizabeth (3) and Catharine (SEP1880.)
In Shediac Parish in 1871, 'Dosity Galang' is age 31, and 'Marey' is age 26. Alfred is age 2 and a daughter is Ada b. MAR1871.
Dosithée Gallant, from the above, was b. NB ca. 1839-42. Possibly in Westmorland Co. I suspect he was the 'Docetta Gallang' b. ca. 1843, s/o Joseph (ca. 1801) and Mary (ca. 1803) 'Gallang' in Shediac Parish in 1851. This appears to be the family found at:
246. JOSEPH7 GALLANT (HELENE6 HACHÉ, MICHEL5, MICHEL4, ANN MARIE CLAIR3 CORMIER, THOMAS2, ROBERT1) was born September 29, 1800 in Barachois, New Brunswick, Canada, and died November 26, 1877 in Aboujagane, New Brunswick, Canada. He married (1) MARIE BONNEVIE January 29, 1822 in Barachois, New Brunswick, Canada, daughter of JOSEPH BONNEVIE and ANASTASIE BABINEAU. She died Bef. 1870. He married (2) MARGUERITE LÉGER May 17, 1870 in Cap-Pelé, NB, daughter of FIDÈLE LÉGER and PÉLAGIE CORMIER.
Children of JOSEPH GALLANT and MARIE BONNEVIE are:
i. FIDELE8 GALLANT, b. Bef. December 19, 1823, Barachois, New Brunswick, Canada; m. OSITHE CORMIER, April 05, 1850, Cap-Pele, New Brunswick, Canada; b. Abt. 1823, Barachois, New Brunswick, Canada.
ii. OSITHE GALLANT, b. April 02, 1826, Barachois, New Brunswick, Canada.
iii. THADDEE GALLANT, b. November 30, 1828, Barachois, New Brunswick, Canada; d. April 22, 1837, Barachois, New Brunswick, Canada.
iv. EUPHEMIE GALLANT, b. August 30, 1831, Barachois, New Brunswick, Canada.
v. URBAIN GALLANT, b. May 17, 1834, Barachois, New Brunswick, Canada; m. DOMITHILDE ROBICHAUD, November 24, 1859, Barachois, New Brunswick, Canada; b. April 03, 1831, Barachois, New Brunswick, Canada.
vi. MARINE GALLANT, b. February 23, 1839, Barachois, New Brunswick, Canada.
vii. DOSITHEE GALLANT, b. April 02, 1842, Barachois, New Brunswick, Canada.
viii. PHILOMENE GALLANT, b. May 22, 1845, Barachois, New Brunswick, Canada.
I am not convinced, however, and suggest the Dosithée Gallant at:
195. Joachim GALLANT (Joachim GALLANT , Francois-Sylvestre GALLANT , François HACHÉ-GALLANT , Michel* HACHÉ-GALLANT , Pierre* ) was born on 9 Jun 1808 in Grande-Digue (Kent) N. B.. He died on 15 Nov 1847 in Grande-Digue (Kent) N. B..
Joachim married Elisabeth POIRIER daughter of Thomas POIRIER and Rosalie BABINEAU on 22 Nov 1830 in Barachois (Westmorland) N. B.. Elisabeth was born on 11 Jul 1813 in Grande-Digue (Kent) N. B.. She died on 5 Apr 1858 in Grande-Digue (Kent) N. B..
Joachim and Elisabeth had the following children:
790 M i Hilaire GALLANT was born on 14 Jan 1832 in Grande-Digue (Kent) N. B..
791 M ii Aime GALLANT was born on 5 Jun 1835 in Grande-Digue (Kent) N. B..
792 M iii Dosithee GALLANT was born on 7 Dec 1839 in Grande-Digue (Kent) N. B..
793 F iv Napoleon GALLANT was born on 11 Jun 1842 in Grande-Digue (Kent) N. B.. She died on 8 Dec 1842 in Grande-Digue (Kent) N. B.. [Notes]
794 F v Rosalie GALLANT was born on 2 Nov 1844 in Grande-Digue (Kent) N. B.
I hope that helps a bit.