The phrase is proverbial. The OED has this under dole:
Portion or lot (in life); fate, destiny: chiefly in proverbial phrase happy man be his dole. c1520–
archaic
Oxford English Dictionary, s.v. “dole (n.1), sense 4,” July 2023, https://doi.org/10.1093/OED/8309182974.
We think of happy as meaning cheerful, but an older sense is fortunate; consider hapless. Brewer's gloss of this phrase verifies that this meaning pertains here:
Happy man be his dole. May his share or lot be that of a happy or fortunate man.
Brewer's Dictionary of Phrase and Fable: Revised and Enlarged, s.v. "Dole". New York: Harper, n.d. pp. 297–298. Retrieved from archive.org 28 December 2023.
Both happy and hapless derive in turn from hap, which the OED defines as follows:
The chance or fortune that falls to a person; (one's) luck, lot; (also) an instance of this. Frequently modified by good (also bad, evil, etc.) and by possessive adjective. c1275–
Oxford English Dictionary, s.v. “hap (n.1), sense 2,” December 2023, https://doi.org/10.1093/OED/6781165414.
These definitions taken together, the proverb equates a person with their lot in life. Hortensio's words suggest "A lucky man is destined to have good fortune" or "Good luck comes to a fortunate man". The tautology acknowledges that whoever wins Bianca will be fortunate. The phrase also expresses a wish for this good luck: the construction be his dole is subjunctive, expressing a desired state, as in Hallowed be thy Name. Further, since Hortensio and Gremio are rivals, the phrase conveys something like "May the best man win".
Shakespeare used this proverb in three other plays. In 1 Henry IV, Falstaff uses it to wish himself and his companions good luck:
Now my masters, happy man be his dole, say I. Every man to his business.
Henry IV Part 1, II.ii.79–80. Accessed at folger.edu 28 December 2023.
In Merry Wives, Slender uses it as a que sera, sera to express his resignation to his possible marriage to Anne despite neither's particularly desiring the union:
If it be my luck, so; if not, happy man be his dole.
The Merry Wives of Windsor, III.iv.65–66. Accessed at folger.edu 28 December 2023.
In The Winter's Tale, Leontes uses it to mean something like "Well, good for you!":
Why, happy man be's dole!
The Winter's Tale, I.ii.204. Accessed at folger.edu 28 December 2023.
These four different instances of the phrase's appearance all help us arrive at a fuller understanding of its meaning. It is simultaneously optimistic, fatalistic, and realistic, expressing a desire for good fortune, recognizing that this good fortune may not be in one's destiny, and signaling an acceptance of this answer the outcome.