may23

Standard png is your best option with transparency included and a smallish file size, albeit almost 3 times the size of a palletised export with approximate file sizes for none palletised being 480kb vs 178kb for palletised.
You could use online or offline image compress apps and you will be able to get the file size down considerably if that is needed.
Settings for imageoptim: https://imageoptim.com/mac
General

Quality

Optimisation Level

Windows users can use: https://pnggauntlet.com/

This looks like a bug and it's the same in V2... Pixel values that are showing as transparent on the 8-bit version have identical colour values to other areas that are not shown as transparent, e.g., the yellow t-shirt, dress and legs...
As others have mentioned, exporting as a 24 bit PNG works as expected and doesn't demonstrate the issue...
If it is essential for the file to remain in 8 bit then you could edit in Photo (if you have access to Photo), make an alpha selection from the channels palette and load it as a pixel selection to a new pixel layer and then give it a white fill and place this layer behind your graphic.
That way you will be able to successfully export it as an 8 bit PNG with background transparency.
This is how you could achieve this (Note: I'm using V2 for the screen recording but the process is identical in V1)... obviously if file size is the key issue  select the layer in the Layers Panel first and use Selection Only when exporting to reduce the exported dimensions of the exported file to that of just the graphic...

kids error.mp4  

I am not sure if it is a bug in itself but the behavior is explained by the fact that Affinity apps do not have separately a property for retaining background transparency and flattening partial transparency when exporting to indexed PNG so partial transparency with matte set to none will remove all pixels with partial transparency.
This can be worked around even without editing the image in Photo by using a Curves adjustment layer where partial transparency has been turned off, like this:

turnoff_partialtransparency.mp4    
 
The resulting indexed PNG with transparent background:
kids error_fixed.png.zip

Thank you so much! 

Go to File > Document setup... Colour Tab and check Transparent Background

That should give you a transparent background for export.