@MiKa If you happen to have time, could you test the range with
a stable 2.7V supply and
a 1mF cap in parallel to the battery?
I am really not happy with the HT7333 solution, since the shelf life of the keyfob will be limited and the quite expensive LIR2032 can easily be damaged by undervoltage.
If the second test works, I could fit 4 220uF 6.3V tantalums next to the LED.
Else I would give the step-up thing another go. There are low profile inductors, but I didn't find any for cheap (shipping) at Aliexpress.