Don't forget that some mumsnetters ARE historians studying the period, legal experts advising Parliaments etc. I am impressed by the level of knowledge - both first-hand and researched - to be found here, with links to verifiable sources.
I think it is inaccurate to teleport the current trans identity back through the decades, for a number of reasons. The most obvious one is that the word 'gender' was not used in the same way in the past. The word 'Transgender' is a relatively recent invention, so it is an anachronism to talk about transgender people in the 1930s.
The term 'transsexual' was in use in the past, and referred to people who underwent sex reassignment surgery to resemble physically the sex they identified with. There was also the possibility of transvestism, which was a less invasive and permanent way to resemble the sex you identified with, or the clothes you liked to wear.
That bears limited resemblance to the transgender movement of recent years, which prioritises the concept of 'gender' over physical sex, so many? most? transwomen retain their male genitalia. They say they are women because of non-physical factors, like believing in a non-material essence of femaleness which they possess, and which is sufficient for them to be women, regardless of biology.
So all transwomen are women, because they say so, not because they have, like transsexuals in the past, had sex reassignment surgery.
The other aspect of transgenderism which didn't exist in the past is the current trans rights movement which has successfully campaigned for males who identify as women to have the rights which used to be specific to biological women; and for the law, medicine, education and everyday language to be altered to reflect reality as defined by the transgender movement.
The term LBGT is anachronistic if applied to the LGB movement prior to approx the 21st century, the T was added [somehow] around 2000. LGB activists prior to that were not LGBTQ++ activists, and shouldn't be referred to as such - if you want to be historically accurate, that is.
So to answer your question: yes, I am arguing that the 21st century group known as 'trans people' did not exist in the the thirties.