• 07.133 early on the fourth-day,

Just so.  A compound word, not a counting word followed by the name of the thing being counted.  Hmmm.

It is an attested compound word, specifically coined by the Society of Friends to mean “Wednesday”.  Beorn did not mean “on a particular day of the week”, but “the fourth day of your travels from here.”  I’m calling a JRRT-original use.

“four, adj. and n.” OED Online, Oxford University Press, June 2017, http://www.oed.com/view/Entry/73969. Accessed 13 September 2017.