Clodius Does this level of double-softening negatively impact extraction quality?
Water make up has no effect on measurable extraction (the objective amount of coffee solids dissolved). Hard water and distilled water extract the same amount of coffee, in the same circumstances.
The hardness & bicarbonate do affect the way that extracted coffee (and other beverages) tastes, particularly bicarbonate.
If you extracted all your coffee with distilled water (very bad idea for most espresso machines), you could then change the flavour with these minerals, by adding them to the cup.
Back to your specific question, Cumbrian Armathwaite/Eden Vale water (sold by Tesco & Waitrose) is on the lowest end of recommended bicarbonate. Without knowing what the Lelit filter you have actually does, it’s hard to say. But I wouldn’t be wanting to drop the bicarbonate any further.
A KH/Alkalinity drop kit can be picked up on Amazon easily and you can use this to test your filtered water, aim for at lease 2 degrees/36ppm alkalinity.