Doesn't look like the ECMWF does.
Perhaps part of it has to do with the fact the weather changes due to eclipses are usually fairly friendly/tame. This GOES 16 satellite loop from the southeast US during the eclipse shows a large percentage of the cumulus buildup dissipating as solar input dropped. And this great meteogram from the Oklahoma Mesonet (this site is Blackwell, Oklahoma in northern Oklahoma, though most sites show some impact) shows the slump in solar radiation very well (the bottom plot) and also temperature and wind speed changes during the early afternoon of August 21, 2017 (very slightly left of the middle of the time period of the graph):
Typically changes like cooling and slackening winds are not conducive to storms, and as the graph shows, are rebound soon after the eclipse end.
It wouldn't seem it's much trouble to include the eclipse... just include a single if statement that checks a date variable holding the next eclipse time. And then, if it is the eclipse, just modify the insolation variable, probably just by the percentage of the sun's area covered up (which there's probably pretty simple equations to calculate?). Just seems most haven't really taken the time, perhaps in part because of the event rarity. Because major operational models are improved during quite regular cycles, including such a rarely-applicable feature like that may end up as more maintenance work than the benefits are worth.
There may not be much in the way of dangerous weather created during an eclipse, but that doesn't mean there is no weather. The next US eclipse in 2024 does offer more potential to be a more weather significant encounter. Most high-end severe weather in the US occurs during spring as shown with:
And there have been a few bouts of severe weather on April 8th, including small bursts in 2013 and 2015, plus the 1998 Alabama F5. Radiation in all of those regions will be quite heavily affected by the 2024 eclipse (coverage > 80% anywhere between Alabama and Nebraska). Would the decrease in energy disrupt severe weather potential later in the day? An interesting question we'll have to see on!
Overall there weren't large impacts during the 2017 eclipse. As mentioned, temperatures dropped for a little while, and a few clouds dissipated. But could the regional disruption of the eclipse introduce subtle wind shifts or temperature changes that alter the longterm pattern, more significantly impacting model forecasts later in time during periods of greater weather variability? An interesting question that would be great to see studied in more detail.