Hi There
I find you are best not to get stuck into 'facts', not to be too rigid, sometimes thinking outside the box gives leads.
'Mary' was just a suggestion, an example, an idea.
An example is my Great Grandmother gave two of her daughters the first name of Mary, one was Mary Bridget and the other was Mary Hannah.
Both girls lived but were never known by their first names.
The lead that myluck has given you is for a Mary Philomena Walsh, not a Philomena Walsh but sometimes these variants can hit the jackpot.
Look for just P as a middle name during searches is another thing to look for.
If she was 'sent' outside her home area, again the 32 counties might need to be looked into for deaths.
If she was sent to an institution for ill health (or variants) and she passed away there, sometimes clerks there would not be fastidious about names.
She could have been given a different name in an institution.
As I'm sure you know, when people went back into the 'system' back then, deaths were not even registered.
You just have to google to see some of the oh so sad reports of the time.
Did a lot of the children pass away ?
Just a thought, MIGHT she have been adopted ?
I have heard stories of where couples whom adopted out a child if there was a lot of ill health in the family, in the hopes that they might 'escape' it ~ possibly not as late as the 1930's but again just a thought.
Anyway, I'm just throwing ideas out here, none fact, just ideas and thoughts

I really admire you trying to find this 'lost' little girl and wish you the best in your endeavors.
Tara