@merrymouse
Quite happy to believe that, but you would need to first acknowledge that two separate groups are involved and that there is a conflict of interests.
I have no problem at all acknowledging that multiple groups have an interest/conflict.
To pick up on the three things that you mentioned in your original post
Prisons, sports, puberty blockers?
Prisons? well its just got completely out of hand now hasn't it ? But we'll come back to that issue last
Sports? The problem we have here (imnsho) is simply solved.
If you (generic) are good at sport and also want to transition then ...
Choose:
Transition Now and Give Up your Sport (as something you do hoping to be a professional/Olympian/ Amateur or School Champion)
OR
Delay Transition and continue to compete in your sport against competitors who share your birth sex.
You CANNOT
Transition and continue to compete in your new gender - this is the option that needs to be removed. (without any bollocks about Testosterone levels etc)
It would be a hard choice for all committed sportspeople but it is a Choice that they can make individually, Sport or Transition, Whichever is more important to them and if they choose sport they can do so knowing that Transition remains an option when Sports careers are done.
Puberty Blockers ?
Ugh. Never took them myself and remain unconvinced that anyone can take blockers for a period of time and (perhaps after a change of heart/mind)stop taking them and have no long term damage.
So ... lets make the minimum age to use Blockers at least 16 ... and yes, doing so would negate the benefits of taking blockers, since people are well in to puberty at 16.
But ... if an individual wants to use blockers at the age of 16 then ok ... because all human beings have autonomy over their own bodies and life decisions. But use of blockers should not be available to anyone who is not in a recognised care pathway, and has not been attending psychotherapy for at least 2 years From the age of 16 - which means blockers can only be accessed by people over the age of 18 (and frankly, I am not sure what the point of a blocker at that age would be ... probably -hopefully- minimal or none).
Now, the Prison thing ... is it fucking stupid or what?
What follows will get me slapped by both sides of this debate but it is my considered opinion (whatever that might mean 
)
WE need to give serious consideration to the introduction of Two Tier 'Gender Recognition Certificates'.
Tier 1 is available to anyone who wishes to use/enter a care pathway which may or may not lead to medical/surgical bodily changes.
Holding a Tier 1 certificate confers NO rights to be sent to a Woman's Prison (or use changing rooms etc used by your target Gender) It simply permits an individual to declare their intent to undertake gender change (for want of a better phrase) and gives access to medical services (Psychotherapy, Hormone Therapy etc)
Tier 2 would only be available to individuals who have completed both social and medical transition (what we refer to as Post-Op). This tier would give access to a Women's Prison if req (and looking at the whole prison question over the last few years, it strikes me that some men make outlandish claims to be women because they think it will gain them an advantage ... so remove all the advantages ... yes, anyone can transition, even a prisoner but if this is a new discovery on the prisoners part then said prisoner will be entitled to a Tier 1 GRC and thus must serve their sentence in a prison that matches their birth sex.
OUCH 
I know that no one likes these sorts of compromises (Whichever side of the debate you may be on) but some kind of compromise must be reached ... maybe (probably) not this but something similar ...
Oh and just to confirm, TRA's do NOT speak for me or on my behalf ... EVER.
Have a good day ...