Can Nisei MK II Be Used During an Ice Encounter?
Yes, but after the runner has a chance to break subroutines, and before any of the unbroken subroutines are resolved.
As pointed out in this answer, Nisei's is a Paid Ability. From the rule book:

Cards can also host counters and tokens. Hosted counters or tokens can be spent, or leave play, without affecting their host. If a
trigger cost requires one or more hosted counters, those counters must
be spent (returned to the token bank) from the card that the ability
appears on. (page 22)
Paid Abilities - Some card abilities have trigger costs that a player must pay before the effect of the ability can be resolved.
These abilities are called paid abilities. A card’s trigger cost is
always listed in its text box before the effect, following the format
“cost: effect.” (page 11, sidebar)

As explained in the FAQ's Timing Structure of a Run diagram (last page - phase 3.1) linked from this answer, a Paid Ability can be triggered during the encounter with the ice - after the runner has a chance to break subroutines, and before unbroken subroutines are resolved. Thus, in the chum example it does not give you any special combo to trigger the net damage, even though you can trigger Nisei before the encounter technically ends. There's a good discussion of this here.