The date depends on whether one is referring to date of completion or date of printing. The map was completed by April 1750 but the 1st edition was not printed until June 1752.
The earliest public presentation of the map would appear to be the 8th of April 1750 when it was presented to the Academie des Sciences in Paris.
Arader Galleries says:
BAUCHE, PHILLIPPE (1700 - 1773) - DE L'ISLE, JOSEPH NICOLAS
(1688-1768). CARTE DE NOUVELLES DECOUVERTES AU NORD DE LA MER DE SUD.
PARIS: G. DELISLE AND P. BUACHE, .
Note the attribution is to Philippe Bauche [sic] AND Joseph-Nicholas de l'Isle. Buache was married to the daughter of the cartographer Guillaume de l'Isle, half-brother of Joseph-Nicholas.
Raremaps.com also gives June 1752, published in Paris. This was the 1st edition. There then arose a dispute between Buache and De L'Isle. Referring to the 1st edition of June 1752,
Fonte's entrance is at roughly 63°N latitude.
In the face of criticism, De L'Isle published another map in September
1752 moving the entrance to 53°N, as the letters had stated. This
caused a rift between De L'Isle and Buache, as the latter stuck to the
initial misreading of the letters.
Thus the University of Hawaii map is the one from which the June 1752 printing made as it shows Fonte's entrance at 63 degrees N latitude, but it is not identical to the one shown on the Raremaps site. Included in the text top centre on the latter are the words
Presentee a l'Academie, dans son Assemblee publique du 8 Avril 1750.
Par Mr. De l'Isle.
Note: 'l'Academie' refers to Académie des sciences
There is yet another copy of the map here, made after the rift between de l'Isle and Buache and with the date 'September 1752'. It is described as
an unrecorded state of Joseph-Nicolas de L'Isle's (Delisle) map of the
Northern Pacific Ocean and Arctic.
and is attributed to 'The De L'Isle family' (which included Buache) and Jacques-Nicolas Bellin.
This is the text in the top left corner of the map above. Source for both images: Geographicus
As the OP noted, there are dates on the map itself. These refer to discoveries and voyages made. The latest date evident is 1747:
Eau de Wager decouv. 1746 et 1747.
This refers to Wager Inlet
...first probed by Christopher Middleton in the early 1740s and
supposedly one of the possible sources of a Northwest Passage.
1747 is also when De L'Isle left Russia for Paris "with a large map collection". Presumably, he and Buache then started work on the copy which they presented on the 8th of April 1750. Perhaps worth noting is that de l'Isle had previously made a map of the Pacific in 1737.