I don't have an actual quote from the Royal Institution, and I doubt that they specifically gave that as a reason in this particular case. This page from the American Institute of Physics biography says that "custom ruled out women lecturers". I concede that this might be a myth, but I don't think your skepticism is justified. The claim that this sort of reason would be as suicidal then as it is now is, I think, patently false. That sort of discrimination, often justified on the grounds of tradition, was pretty common in the early 20th century.
This is a period when women could not receive a degree at Cambridge, even though they could sit for the Tripos. When Hertha Ayrton was nominated to the Royal Society in 1902 (the first woman to be nominated), the nomination was rejected explicitly because she was a married woman. See here. From the Royal Society's response:
We are of opinion that married women are not eligible as Fellows of the Royal Society. Whether the Charters admit of the election of unmarried women appears to us to be very doubtful.
The relevant charters were only amended in the 1940s.
Today is Ada Lovelace Day, when STEM enthusiasts highlight the work of modern and historical women scientists, engineers, and mathematicians. If you run a blog, you may want to participate by posting about a woman in a STEM field whom you admire. But I'd love to have people share women scientists/mathematicians/authors in the comments that they think we could all stand to read more about.