What temp was the fz while you were carbonating?I purged first then filled and sealed in my swing top bottles right away
When I bottle from a keg (using a beer gun), I like to have the beer around 2oC at 15 PSI. Then vent it to around 3 PSI as you did. Bottles at 2oC too. You’ll lose some carbonation during the bottling, in fact you want to, to ensure you’re capping on foam to reduce O2.
I find that gives me the carbonation I want when I pour.
Btw, you can’t trust the carbonation if the bottle stays cold after filling, you need to warm it up then cool again to really see its carbonation.