Following all the restrictions given in the question, I propose using letters of the writing system as names for days of the week.
If your society is literate and the number of symbols in the writing system is relatively small, the canonical order of letters is going to be very well known.
I don't know of any natural languages that actually do this, although the heavenly stems were historically used to label days of the week based on which sacrifices and rituals were supposed to be performed when. So, the heavenly stems form a small closed system with additional meaning besides designating a day of the week.
Tying the symbols-as-days-of-week to a religion, current or historical, is probably a good way to make the system plausible in your conworld.
So, you specifically mentioned not wanting to name days after numbers, but some very widely spoken languages do exactly this.
I think you can make the 12-day week in two halves thing make sense by having a duodecimal number system with six as a sub-base. In a system like this, some or all the numbers in the 7 to 12 range would be derived from the words for 1 to 6.
In Mandarin Chinese, the days of the week are simply numbered, with Sunday being an exception to the pattern.
In addition, some number systems have a sub-base.
For instance, here is a numeral system developed for an Inuit language with a vigesimal number system by speakers of the language. The number system has a sub-base of 5, which is reflected in the forms of the numerals.
Here is the table in the article reproduced below. The system is not completely regular, but many numbers in the
1-20 range, particularly those congruent to 2 or 3 mod 5 are transparently derived from a small multiple of 5 followed by the word for 2 or 3. Some numerals have irregular forms or are represented subtractively like 9, 14, 19.
0 1 2 3 4
- atausiq malġuk piŋasut sisamat
5 6 7 8 9
tallimat itchaksrat tallimat malġuk tallimat piŋasut quliŋuġutaiḷaq
10 11 12 13 14
qulit qulit atausiq qulit malġuk qulit piŋasut akimiaġutaiḷaq
15 16 17 18 19
akimiaq akimiaq atausiq akimiaq malġuk akimiaq piŋasut iñuiññaŋŋutaiḷaq