Hello,
You just have one single product in your cart.
So there is just one single warehouse for the shipping selection ; and where there is just one warehouse the name of that warehouse is not displayed in the page.
Afterwards, every shipping plugin without a warehouse restriction will be available for every warehouses.
That's why, with the current screenshots you gave us, I don't see a problem, all looks normal.
If you just want to have your special shipping method, you should also add warehouse restriction to your other shipping methods, so they won't be available for all warehouses.
Regards,