So I dont really know this TEK, but maybe I can give an answer.
You say it is told to make 3x water extractions, meaning that you heat it up in acidic water / filter it / repeat for a total of 3x.
If you do it like this, then after 3x every regular bark will be depleted and you will have all goodies out of it.
So if the TEK is that way, you can completely discard your bark, because all the actives will be dissolved in the acidic cooking water.
Now you also talk about a first boiling instead of a heat bath. Sounds like you mean 2 different steps.
Normally I would understand that you want to ask if you should EITHER boil them for the acidic cook OR if you just should place them in a heat bath for the acidic cook.
But how do you mean *Boil before the heat bath*?
If it is this way that you want to know whether do a heat bath or directly boil them, I would say BOIL THEM ;D
Normally greater heat should speed up the extraction. I recommend a temperature of 75 - 85 °C.
But now I dont know what you mean when you really want to do 1 Boiling AND 1 heat bath.
Where should be the boiling and where the heat bath?
What is not described in the TEK is that you simply LEAVE the bark inside of your acidic soup. It simply tells you not to separate them and just proceed as if there bark would not be there.
Actually ... I think this very strange, there is simply no need to leave the bark in. After 3x the bark will be definetly empty, if you heated it up to 75°C+ and did each cook for at least 30 minutes or maybe even 60.
I always remove the bark ... what you dont need should be removed ... so ... but it seems that the TEK still works this way, you can decide what you want to do on your own.
PS: Maybe I can think of the reason why the bark should stay in. Inside the bark there is still some little amount of water, if you separate both. You will never be able so squeeze 100 % of water out of the bark. Especially when using powdered bark. At least for SHREDDED bark you will always have 50 ml of water inside of 100 g bark left, even when squeezing pretty hard. So maybe the TEK is to also recover this last bit, but actually I still remove the bark and get practically maxed yield.