You seem to be referring to the difference in time at which the delay shuts the output off but that should be of no consequence as long as it's less than a second (according to your first post).

Yes, my bad.

On topic of AO3400A, what do you think about that? It's not rated for 1.8V Vgs, but for 2.5V.
Also I have found Si2312BDS that is rated for 1.8V and available in SOT23. Will that work? If not then can you recommend something from JLCPCBs part library?
Also there is another candidate AO3416 that has parameters similar to AO6408, and is ESD protected as oposed to AO3400A. I think that will work the best

 

The MOSFET characteristics are not critical, so any of those should work.
A higher Vgs(th) will just produce a shorter delay, which can be adjusted by increasing the gate resistor to ground value.