This doesn't address the issue of LEO not indicating count vs. non-count nouns in English, but in answer to your question of whether linen is a countable noun in AE, the answer is yes, depending on how you want to use it.
Typically, both M-W and AHD will indicate specifically if a noun not countable. In the case of "linen," neither specifies that. Their silence in that regard indicates that it is countable (or at least not non-countable).
However, their further entries show that it can be treated as non-countable.
The main entry for linen from M-W indicates a non-count usage:
Definition of linen (Entry 2 of 2)
2 : clothing or household articles made of linen cloth or similar fabric
However, directly on the page is reference to an additional usage:
Phrases Related to linen
Definition of bed linen(s)
: sheets and pillowcases for a bed
To me, "(s)" indicates that it may be considered both count and non-count.
AHD is a bit more explicit in this regard:
2. also linens Articles or garments, such as sheets, tablecloths, or underwear, formerly made of linen and now usually made of other fabrics, especially cotton.
Again, either usage is acceptable in AE.