This is another complex passage with an intervening parenthetical comment. The verb in the last main clause, become, is not indicative but subjunctive; and it is conjoined with the preceding ellipted predicate (thus “and there build up unto themselves a kingdom … and they become sufficiently strong to contend with the tribes of the people”). The change to became in the 1906 LDS edition could have been intentional, especially if this final clause was interpreted as being conjoined with the immediately preceding until-clause (“until they were joined by dissenters … and they became sufficiently strong”). In this until-clause, the verb phrase were joined is grammatically in the past tense, yet it is a conditional subjunctive. The use of became in the 1906 LDS edition could also be viewed as a conditional subjunctive.
Under either interpretation, this passage is difficult to process. But since either reading will work, the critical text will follow the reading of the earliest textual sources, 𝓟 and the 1830 edition: “and they become sufficiently strong”. Since these two firsthand copies of 𝓞 agree here, 𝓞 itself very likely read become. For a list of other cases where become and became have been mixed up in the transmission of the text, see under Alma 13:9.
Summary: Accept in 3 Nephi 7:12 the use of the present-tense subjunctive become, the reading of the earliest text; either the present-tense subjunctive become or the past-tense subjunctive became (the reading in the 1906 LDS edition) will work here, so the critical text will follow the earliest extant reading.