Credible Threat in the Perfect Bayesian Equilibrium
I am going to assume that $F$ and $G$ are continuous distribution functions (I.e. players are playing properly mixed strategies). If this is not the case there is not enough information to answer your question. Given that assumption, player 1 and 2 must be indifferent among their actions in game 1 given the strategies of the other player. 

In the second game, you are assuming that the node given by $x,k$ is never reached (since $j$ and $k$ give the same payoff and when indifferent, player 1 always choose $j$). 

Under that behavioral assumption (which must be verified to be consistent with equilibrium), player 2 must believe she is at the node "$y$" with probability 1. Since Player 2's beliefs have changed with respect to game 1, the mixed strategy given by $G$ is, in general, no longer optimal (unless $a$ and $b$ give the same payoff regardless of the action of player 1). 

In short, player 2 will, in general, not use the same strategy in the first and the second game; so threatening to use $G$ is not credible. 

Btw all threats are credible in a PBE since strategies are best responses given some beliefs that are Bayesian.