After some searching in all kinds of esoteric ways, I was able to find a NYT image of Ernest Blumenschein's portrait of Lionel Barrymore's family circa 1909 in Paris! Lionel is tending toward the "aldermanesque girth" sister Ethel remarked him having when she saw him either just after or in Paris.
I have not been able to find a picture of the painting itself yet, but as it was shown at the Art Institute of Chicago as well, I may be able to. This was being shown at the Pennsylvania Academy of Fine Arts in the Jan-Mar 1909 exhibition. Note the info below on Blumenschein's relationship with LB:
According to this excerpt, Mary was the firstborn, while Ethel was the second. But according to the NYTimes, August 30, 1908 (and it would make sense given Ethel Barrymore was subsidizing their Paris life while Lionel studied art), Ethel would have been the child unless she died very, very early after being born (some accounts indicate one of the girls only lived a few months):
If one of his daughters died while both Blumenschein and Barrymore were studying in Paris, it would have been between 1907/8 and about late 1910. If so, it was likely Ethel II who was represented in the above portrait. For some reason, this portrait was also chosen for satirizing by "the Fakirs", which you can see in the NYT of 17 Apr 1910.
Generally little is still known about the daughters of Doris and Lionel. I will keep plugging away.
Barrymore remained on friendly terms with Blumenschein after they both left Paris: I found this letter on the Smithsonian Institution's page for the painter:
The letter referenced in the google book sample may well reside at the University of Iowa, as Ellis Parker Butler was an Iowa native. The library has a goodly collection of his letters, including some from Blumenschein.