version Compositor
TheLabCat's comment in images :) just ... I don't think VSE can handle this blur task, so ... version in compositor.
Originally I used Math node > Less than 0.5, but edge was jagged (without antialiasing) so I used ColorRamp node instead.
version Sequencer
For VSE you would have to use Mask (in a current version there is not an one click solution like in other apps, but on other hand with this system you have open a gate to endless and much more complex and animated masks.
It will be hard to create round corner in native blender's Mask system with rounded corners (seen under Paint Editor).
Would be probably more easier to create a "mask" object in 3Dviewport (or use bw image imported as a strip).
Here is example with mask from Scene ...
You already have one scene with VSE setup. Create another Scene in the same file and in 3Dviewport set Camera to Orthographic, Add a Plane at location you want to see Small Screen through Camera, add Solidify+Bevel modifiers, set white material and black world ...
Left camera is the mask for SmallScreen, right camera with Plane is duplicated + one more Solidify modifier. It will works like a stroke thickness for Colour strip :)
So you should see something like this ...

In your VSE scene add two Scene strips (containing your masks) and under properties choose corresponding Camera object. Strips can be placed whenever you want (or hide them). We need them just for strip modifiers.
For Small Screen add Mask Modifier > Strip and choose Scene strip with mask done at the first camera. For a stroke add Color strip and ad Mask modifier > Strip > Scene with camera two ...

File contains both version ...

To be more flexible you can create a rounded mask for entire screen, and under VSE apply mask to "small" screen at size of full screen, group strips and transform (scale, translate) whole group (image with stroke).