This mode has nothing to do with the Blender tweaking tool.
Here is how it works: you select the island you need and press Fit To Trim. Then open the operator settings (the black panel in the bottom-left corner of the screen) and adjust them as required.
By default, the operator works in Fit: Tweak mode. This is done so you can adjust the operator and quickly place an island into a trim the way you need, without changing the trim’s own properties. It is a simple and fast mode for straightforward cases.
However, if you want to place many islands and you know exactly how each trim should work, you should configure the trim properties first and then switch the operator to Fit: Exact mode. This is what I showed in the previous screenshot.
Regarding whether you need to do this again — no, you don’t. Operator settings are preserved for the duration of the Blender session until you change them manually.
What I described above is the basic interaction model for any Blender operator. Spend a bit of time getting familiar with it, and your workflow will become much faster.