It appears there was a child born in Paris to them, named Mary, who seemingly was born and died in 1906, the first year they were in Paris. Peters mentions the survey by Daniel Blum LB answered in "the 1950s", in which he said he and Doris had two daughters, neither of whom survived infancy. In the same bio, she says John, Lionel's nephew (I presume she means John Jr) said a daughter died in the flu epidemic of 1917-18. With that, I set forth and starting digging--and oh, what a dig.
It is clear Ethel Barrymore II was indeed born, in Paris, in 1908 and died in 1910 in New York. She was with Lionel and Doris when they arrived in New York August of 1909, and was about a year old, according to news accounts. I have gathered from the PHENOMENAL fultonhistory.com site various New York newspaper accounts of this period, including several which mention Ethel II's death.
On Mary Barrymore, born in the US, I was able to find out the following:
BARRYMORE KIN DEAD
Mary Barrymore, aged one and a half
years, the infant daughter of Mr. and
Mrs.Lionel Barrymore and niece of Ethel
Barrymore, died at the family residence
in Garden City, Long Island, Monday
from pneumonia. The child had been ill
five days prior to her death.
(excerpted from The New York Clipper, March 21, 1917, page six, Vaudeville section, fultonhistory.com)