You probably won't notice the difference. Pi Hat DAC mostly were using similar chipset just like wiim mini. I came from experimenting myself in this journey. computer 3.5mm > apple mac optical out > pi + hat > pi usb + external dac > wiim mini optical + external dac.
Just my personal experiences. I would tell you to leave your pi solution because I've spent so much time and $ on it already. It's just....... so so....
As a side note.. I am not an expert on electronic design. But I think you are heading to the right direction but not entirely correct.
Most of cheap dacs such as cd player, computer mother board, iphone internal dac, wiim mini.. cheap pi hat dac.. they are all "somewhat" reached to the engineer excellency as you mentioned " if they are engineered reasonable well they should produce the same analogue output".
However! the missing link isn't just analog output stage. there are more to it. Otherwise how come decent priced DAC chip, almost every manufacture uses it. how come everyone came up with different sound signature? Keep in mind, even same company but just a tiny different production line produces some differences.
It should really separated as in "digital output" and "analog output" stages. Something like SMSL, Topping, they would apply some digital filters to silent some background noise first. Then they would apply their own "EQ" filters before audio stream goes down to analog output. Although analog output is extremely important when people talk about delta-sigma, R2R.. That's just further filtering and coloring to the audio stream itself.
It's really not as simple as "all dacs were alike" or "if they are engineered reasonable well they should produce the same analogue output".. That's just false assumptions.
So the answer to your question after this huge circle I am making. NO.. between wiim mini and pi + hat you won't hear a difference. But with wiim mini optical out to an external dac, you will hear A LOT differences. That's because your wiim mini is no longer "coloring" the music but the expensive external DAC is.
What you are comparing is like.. "would I hear a difference from a $30 headphone vs my apple headphone came with my iphone for free?".. Obviously there are differences but the difference is so minimal you won't feel a difference. However, when you attach an external dac is like you upgraded your headphone from a free one to a Sennheiser hd 6xx.. all I can say is.. If you can't hear a difference... then you are better off to save $. Audio equipments are waste of your hard earned money.