The printer’s manuscript has “the last record”, but the 1830 edition has “this last record”. The 1908 RLDS edition restored the reading of 𝓟. However, it is apparent from the context that the word last must mean ‘last mentioned’, not the final record of the Nephites. Moroni, not Nephi, kept the final (or last) record of the Nephites.
Moreover, transmission errors strongly suggest that the more frequent tendency was to accidentally replace this with the rather than the other way around (see, for instance, under Helaman 14:20 the extensive list of mix-ups between the and this in the early transmission of the text). In particular, there are four clear instances where scribe 2 of 𝓟 wrote the instead of the correct this, including one that involves record:
In this later passage, Oliver Cowdery corrected “in the record” to “in this record” when he proofed 𝓟 against 𝓞. And his correction agrees with the reading of the 1830 edition, which was set from 𝓞. Scribe 2 of 𝓟 probably wrote “in the record” in Mormon 7:8 because of the phrase “in the record” that follows: “but also in the record which shall come unto the Gentiles from the Jews”. Similarly, here in 4 Nephi 1:19 scribe 2 may have written the record in place of this record because of the plates in the immediately following “and he kept it upon the plates of Nephi”. For another case where scribe 2 of 𝓟 seems to have replaced an original this with the, see under Mormon 1:16. On the other hand, there are no clear cases where the 1830 typesetter replaced an original the with this; for discussion of this point, see under Helaman 14:21 (and earlier under Helaman 14:20).
Summary: Maintain “this last record” in 4 Nephi 1:19, the 1830 reading; here “this last record” means ‘this last-mentioned record’, not ‘the final Nephite record’; there is also considerable manuscript evidence that scribe 2 of 𝓟 tended to replace this with the, including one other case where he replaced this record with the record.