Are there any particular laws surrounding the toilets in Japan for
transgender people?
I'll be first to answer the actual bathroom law question. Firstly, there is no bathroom law in particular, but it falls under the jurisdiction of trespassing law (住居侵入等,
article 130).
Here's what it states:
Article 130 of the Criminal Code.
A person who, without justifiable cause, enters a person's residence or a residence, building or ship guarded by a person, or fails to leave these places despite being requested to do so, shall be punished with imprisonment for a term not exceeding three years or with a fine not exceeding 100,000 yen.
Thus if you were to obtain permission from the owner of the building where the toilet resides in, you will be able to use whichever toilet regardless of your presentation.
However, another law comes into play if a third party sees you and reports you to the police, which falls under "Violation of the Anti-Nuisance Ordinance (迷惑防止条例違反), and states:
Nuisance prevention ordinances are ordinances set by each prefecture
and other local authorities prohibiting nuisance behaviour towards the
public.
In many municipalities, the official name is 'Ordinance on the Prevention of Violent Delinquent Acts, etc. that Cause Extreme Disturbance to the Public' (e.g. Tokyo Metropolitan Government and Osaka Prefectural Government).
Basically, a nuisance act committed in a public place or in a public vehicle is a violation of the nuisance prevention ordinance.
...and transgender individuals have been arrested for it in the past.
Thus, the only legal recommendation is to use the gender neutral toilet (多目的トイレ).
Short list of transgender individuals who have been arrested for toilet use in the past 2 years (by request):
There are a ton more articles, but as explained, they are rarely ever tagged 'transgender' due to public and lawful perception.
Bonus: "Gender neutral toilet removed as it breeds crime"