BvS is the first time we see Kryptonite referenced in the new DC Extended Universe so there are no hard and fast rules in play here so we can only interpret what we see.
Previous movie versions/interpretations of Superman & Kryptonite or appearances in other media do not apply here
The first time Superman is exposed to it is in the form of a gas fired by Batman..
...and while this weakens him it's only a temporary effect. It's sort of like a suckerpunch...sure it'll hurt for a while but you'll soon shake it off, especially when in a fight for your life.
A second exposure again puts Superman down enough for Batman to, again temporarily, gain the upper hand.
When Batman brings the spear into play, it's mere presence does not seem to affect Superman further but it (perhaps in combination with the gas) allows Batman to cut/penetrate Superman's skin
Note...Batman is planning on stabbing Superman here. If Kryptonite was completely incapacitating Batman could just leave it on Superman's chest and walk away.
Kryptonite (in this version at least) isn't instantly lethal...it's more pervasive...kind of like a radioactive metal. It's going to kill you but slowly and from inside.
Obviously close exposure to a chunk or weaponised Kryptonite (like the gas) temporarily weakens Superman..it doesn't instantly kill or suddenly wipe away all his powers. It just reduces them.
When you see him pick up the spear it's clear he's still weak...he can barely fly...but he's summoning all his strength and abilities to do it.