If you have a 110 bolt circle diameter, the smallest chainring you can fit for the middle would be 34, not 32. The one you cite should work. You however should ask the seller of that chainring if the ramps and pins are correct for using a 44t as an outer ring.

If the ramps and pins are not compatible with each other between your middle and large chainrings, the shifting won't be as smooth. In the 1960s, 70s, and 80s, nobody had ramps and pins, shifting was ok. So it is not a big problem if your ramps and pins are not compatable, it is just a less smooth shift and you can't be pedaling hard when you shift.

I can't figure out why the downshift onto the 24t chainring only works well if you are on one of the two smallest cogs on the cassette. Perhaps it is a front derailleur issue? I am using a derailleur from the 1980s, seems to work best for me.