(Note: I'll be using regular chessboard diagrams to illustrate this.)

So,

the only way a lolcat can be captured is if a piece transmogrifies into a knight, exactly one knight move away from it.

Here's a nice diagram:

This diagram shows what a piece landing on each square transmogrifies into (assuming it is moving towards the lolcat). Note the board is infinite so this goes in all directions.

Alright, now we can attack the puzzle.

An optimum setup for Black would be to place two lolcats so that whenever one is captured, the other one can capture back (considering the capturing piece is now a lolcat.) The second lolcat can't directly defend the first, because the captured piece would transmogrify into a lolcat, then immediately get another move and capture the other lolcat. Placing lolcats on d1 and f9 seems to do the trick...

BUT there's a problem! Once capturing and transmogrifying into a lolcat, the new lolcat immediately gets another move. In this case, the new lolcat can move to d8, transmogrify into a knight, and capture the second lolcat. SO, we need to find a setup where Black has a connected network of lolcats where, if one lolcat is captured, another one can recapture AFTER the new lolcat makes a move.

Basically, we need to use a lolcat as bait.

Unfortunately, I have no idea how to do this, but it's a start. Will probably update soon.

(Note: I'll be using regular chessboard diagrams to illustrate this.)

So,

the only way a lolcat can be captured is if a piece transmogrifies into a knight, exactly one knight move away from it.

Here's a nice diagram:

This diagram shows what a piece landing on each square transmogrifies into (assuming it is moving towards the lolcat). Note the board is infinite so this goes in all directions.

Alright, now we can attack the puzzle.

An optimum setup for Black would be to place two lolcats so that whenever one is captured, the other one can capture back (considering the capturing piece is now a lolcat.) The second lolcat can't directly defend the first, because the captured piece would transmogrify into a lolcat, then immediately get another move and capture the other lolcat. Placing lolcats on d1 and f9 seems to do the trick...

BUT there's a problem! Once capturing and transmogrifying into a lolcat, the new lolcat immediately gets another move. In this case, the new lolcat can move to d8, transmogrify into a knight, and capture the second lolcat. SO, we need to find a setup where Black has a connected network of lolcats where, if one lolcat is captured, another one can recapture AFTER the new lolcat makes a move.

Basically, we need to use a lolcat as bait.

Alright, so I'm pretty sure it's impossible. Trapping a queen is difficult in chess. This is trapping a queen that can also move like an infinite knight on an infinite board that has a ton of extra moves to screw you over.

I'm not selling this well, am I?