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Anyone able to narrow this down to genus or further?
Only 10 genera known to exist here.
Which can be ruled out?

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List of known Anthomyiidae species in Iceland :

Alliopsis sepiella (Zetterstedt, 1845)

Botanophila betarum (Lintner, 1883)
Botanophila fugax (Meigen, 1826)
Botanophila profuga (Stein, 1916)
Botanophila rubrigena (Schnabl, 1915)
Botanophila silvatica (Robineau-Desvoidy, 1830)
Botanophila tuxeni (Ringdahl, 1953)

Delia angustifrons (Meigen, 1826)
Delia echinata (Séguy, 1923)
Delia fabricii (Holmgren, 1872)
Delia platura (Meigen, 1826)
Delia radicum (Linnaeus, 1758)
Delia setigera (Stein, 1920)

Egle lyneborgi Ackland & Griffiths, 2003
Egle minuta (Meigen, 1826)
Egle pilitibia (Ringdahl, 1918)

Fucellia fucorum (Fallén, 1819)
Fucellia maritima (Haliday, 1838)

Hydrophoria albiceps (Meigen, 1826)

Lasiomma picipes (Meigen, 1826)
Lasiomma seminiditum (Zetterstedt, 1845)

Pegomya bicolor (Wiedemann, 1817)
Pegomya fulgens (Meigen, 1826)
Pegomya furva Ringdahl, 1938
Pegomya icterica (Holmgren, 1873)
Pegomya notabilis (Zetterstedt, 1846)

Pegoplata infirma (Meigen, 1826)

Zaphne ambigua (Fallén, 1823)
Zaphne brunneifrons (Zetterstedt, 1838)
Zaphne divisa (Meigen, 1826)
Zaphne frontata (Zetterstedt, 1838)
Zaphne subarctica (Ringdahl, 1918)

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It's not Fucellia, a genus with dichoptic males. In my experience Egle fly early in the year. The arista of Hydrophoria is plumose. Pegomya bicolor is bicolored; most species of Pegomya are lighter in color than your fly. Delia platura has about 20 short posteroventral bristles on the hind tibia. I don't know what your fly is.

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Thanks John!
According to this
https://diptera-in-beeld.nl/Ref-Key%20British%20Anthomyiidae.pdf

Perhaps we can also say

Delia = Eyes
separated by almost the width of postpedicel.

Lasiomma = Eyes with distinct pubescence

The image on BOLD systems for Alliopsis sepiella and Pegoplata infirma looks to be an incorrect match for the face on both... yellow halteres on the former too.
https://v3.boldsystems.org/index.php/Taxbrowser_Taxonpage?taxid=879093

This would leave options of

Botanophila

or

Zaphne

From the list

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The genus name Lasiomma means "hairy eye", but in fact some species have bare eyes.

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Ah right, yes, missed that part in the key - L.picipes has bare eyes and ... :
"mid tibia with 1 ad, 1 pd and 2 pv setae (not in a true posterior position, see couplet 45). About 4-5 mm long, mainly blackish species, genal setae in more than one row, 3 posthumeral setae, costal spines at distal break hardly discernible form the spinules between the costal breaks, scutum with 2 presutural grey dusted vittae between acrostichal and dorsocentral rows. Hind tibia with 8–11 ad setae. All tibial setae rather short. Hind femur with av and pv setae"