Most of this is disjointed as I kept looking while Rootschat was not available. I updated my browser and Rootschat was the only site I could not access, now resolved.
This morning, I went for a drive, down the tiny roads either side of Crwbin and found nothing. Then I called at the library and looked at an OS Map. Lan is still present and named as a hamlet of houses. There is a property called Van Farm very close to the hamlet of Fan I found on the 1889 map, off the B4309, northwest of Meinciau. It is on the other side of the road to the present quarry. Looking at Street View, it is a large farm complex, though probably not dating back to 1840. From the aerial view, there are more houses I could not see from the road. When I click on those houses, it says they are in Llangyndeyrn, not Meinciau. On the OS map, Llangyndeyrn was a large district that took in most of the limekilns and quarries.
I have found two places called Plas-bâch. https://maps.nls.uk/view/102179738Carmarthenshire XLVI.SE (includes: Kidwelly; Llandyfaelog; Llangyndeyrn.)
Revised: 1905 . Published: 1907. Bottom left corner. It is the opposite corner to Fan and may not be in Llangyndeyrn.
But there is another just outside Llangyndeyrn. https://maps.nls.uk/view/102179744 - Carmarthenshire XLVII.NW (includes: Llangyndeyrn.) Revised: 1905 . Published: 1907.
Top left corner. Llangyndeyrn is on the left edge just above halfway. Looking on Google Maps, it could still be standing and now a farm.
https://newspapers.library.wales/view/4347394/4347397/35/The Welshman . 23rd May 1851
THE COST OF LIME. To the Editor of the Welshman. Sir, Will you allow me to ask through your columns what is the reason that lime at the Van Issa and Mynydd-y-garreg Quarries, situate in the parishes of Llangendeirne and Kidwelly, has risen in price nearly double to what it was in years of Protection?
https://newspapers.library.wales/view/3582375/3582377/22/The Carmarthen Weekly Reporter (Supplement) . 26th March 1897
fatal accident took place at Van-issa Quarry, Llangendeime, about 10 a.m. David Bowen (38), a lime burner and quarryman, living at Blaenpant, was working at a portion of the quarry rented from Mr Fitzwilliams, Adpar House, Newcastle-Emlyn.
https://newspapers.library.wales/view/4352090/4352094/11/The Welshman . 24th April 1863
It talks of constructing a railway line from Kidwelly and is an interesting read, about the wealthy owners wanting more. ‘One is a revival of these mineral branches, under a new company, the other is to make a line from Kidwelly to Van-issa and Van-ucha, and to Gorsgoch, at the top of the Llanelly Railway, for the purpose, it is stated, of opening all the lime and coal in the valley.
From that, it is clear Van is a quarry with issa and ucha being lower and upper. If you follow parts of the line on the 1907 map it heads up the valley towards the mining area near Fan, then probably sweeps around to the area of Lan. The issue is not knowing where Gorsgoch is, as I could not find it on Genuki, but from newspapers in Jan/Feb 1880, it seems ‘THE GORSGOCH COLLIERY and FIREBRICK 1 WORKS, situate at MYNYDD MAWR, near LLANELLY’ which then links lime to coal and Llanelly, then Swansea.
https://newspapers.library.wales/view/4355476/4355480/19/The Welshman . 16th July 1875
LOT 3. All those two desirable Farms, called "LAN & WERN," With suitable Farm-house and Out-buildings, and 122a. 3r. 3p., or thereabouts, of capital Arable, Pasture, and Meadow Land, in the occupation of Mr John Thomas, as yearly tenant; together with the Lime Rocks and Kilns, called Van-ucha, and the Lime-stone Quarry… The Property is well supplied with water, it is intersected by capital roads, in close proximity to the celebrated Van Lime-kilns, and within easy distances of several well-known Collieries. It would seem, Lan was a farm not a hamlet.