CustomScrollview with TabBarView that overflows

So I'm having some difficulty using the following Setup:

CustomScrollView(
     slivers: [
          SliverAppBar(...),
          SliverToBoxAdapter(SomeRandomWidget),
          SliverPersistentHeader(TabBar),
          SliverFillRemaining(TabBarView)
     ])

So the TabBarView contains a Column on one tab, that overflows the screen. The issue is, that even though SliverFillRemaining is obviously not the right choice for the overflowing Widget but sadly as TabBarView looses the hasSize property of its children, I don't know what other Widget I could use to get the layout going. The goal is to achieve something like this (shows the current overflow):

EDIT: Here is a working example on DartPad: https://dartpad.dev/bda4cc5fd2aea292310fe05daa440760