The original information from Wikipedia is suspect. Also, no post office would ever be opened "early" before authorization. They would not have any stamps, equipment, etc. Also, the Jim Forte website is a secondary (or tertiary) source. When encountering conflicts between his site and official government records, one should side with the records. So let's go there.
You have found the site location microfilms. They are good source for locating offices AND to use the maps from the records of the other nearby towns to cross-check. I see Lucerne's site aplication was dated March 1, 1903 at the bottom, and apparently received back in Washington, DC on March 19, 1903.
Perhaps more important would be return of the postmaster's bond and oath documents, not available online. However, searching the "Postal Bulletins", issue 7086, dated June 1, 1903, notes the establishment of Lucerne effective March 28, 1903.
It would be hard to quibble with that establishment date and merely a transcription error on Jim's site.
As a tangent to explore further, the site form lists the Lucerne as a "mining camp", which sent me to Chester M. Smith, Jr.'s article "Summer and Winter Resort Post Offices" in the July 2000 issue of the "American Philatelist". He lists Lucerne as a summer office, 1909-1930 (an interesting match to Wikipedia). Postal Bulletin 9067, dated November 23, 1909 lists a seasonal closing:
And Postal Bulletin 9173, dated March 31, 1910 lists the reopening in 1910:
Smith notes the Postal Bulletins were very irregular about publishing the annual openings and closings of seasonal offices. I did not look for any other announcements.
This leads me to believe Lucerne was a regular PO 1903-1909 (reinforced by the January 1, 1907 "Postal Guide" listing Lucerne as a full, non-seasonal post office), a summer PO 1909-1930 reinforced by Lucerne marked as a summer office in the July 1, 1930 "Postal Guide", and unmarked in the July 1, 1931 edition), a regular PO again 1930-1956, and lastly a Station of Holden 1956-1957.
The 1956 and 1957 transitions from the Jim Forte post office list lead back to the Postal Bulletins and what turned out to be a far deeper rabbit-hole than I expected, but an incredibly complex comedy of errors too amazing not to share!
Five postings in the Postal Bulletin to get Lucerne's switch to a contract station in 1956 (and I may have missed one regarding June 30 vs July 2):
PB 19946, dated May 17, 1956
PB 19954, dated June 14, 1956
PB 19967, dated July 31, 1956
PB19975, dated August 28, 1956, and
PB 19976, dated August 30, 1956, respectively.
And one Postal Bulletin announcement to finally close Lucerne in 1957:
Posted 03/01/2023 2:00 pm
