Lord of the Rings / Phases of the Moon II

In his fantasy masterpiece Lord of the Rings author J.R.R. Tolkein was in general consistent with real world astronomy when mentioning the phases of the moon. In a previous post A Peculiar Moon in Lord of the Rings I noted a couple of exceptions. Alas, I'm in the middle of rereading it for the zillionth time and found another glitch, this time in volume I, Fellowship of the Ring. This webpage Moon Phases in The Lord of the Rings kindly reproduces the offending passage and also sets out the relevant facts about the phases of the moon, but did not point out the inconsistency.

Above him was a black starry sky. Suddenly a pale light appeared over the crown of Weathertop behind him. The waxing moon was climbing slowly above the hill that overshadowed them, and the stars above the hill-top faded. ... ‘Look!’ said Merry. ‘The Moon is rising: it must be getting late.’

However, a waxing moon rises during the daylight hours, not at night!
Oddly enough, just a few pages earlier, while describing events taking place two nights previously, Tolkein mentioned the phase of moon accurately. “The moon was waxing, and in the early night-hours a cold grey light lay on the land.” That is correct: a waxing moon rises during the daylight hours and in the early night-hours it is still above the horizon.

Here's a speculative attempt to figure out in some detail the state of the moon the day of the incident at Weathertop - October 6 S.R. 1418. In appendix D, it says the our New Year's Day corresponds more or less to the Shire January 9. That means that the full moon - which they saw on January 8th S.R. 1419 when the Company reached Hollin - would correspond to our December 31st. There was a full moon in London on Dec 31, 1933, which can be used as a reference point. https://www.timeanddate.com/moon/phases/uk/london?year=1933

Also our 26th of September is probably a good approximation for October 6 S.R. - the date of the incident at Weathertop.

On September 26, 1933 the moon was half full and waxing https://www.timeanddate.com/moon/uk/london?month=9&year=1933.

6:52 sunrise 26th September
15:39 moonrise
18:51 sunset
22:25 moonset
6:54 sunrise 27th September, the next morning
16:23 moonrise the next afternoon

Daylight savings time was in effect during September in 1933 in England - not sure about Middle Earth on the corresponding date - but the intervals between sunrise/sunset/moonrise/moonset would not be effected.

So by that rationale, the moon would have never been above the horizon during the night of the incident at Weathertop.