Fixing tear in vinyl floor
There is no way to fix that.  Just replace the bathroom flooring.  A small area won't be expensive if you DIY.  

Maybe wait until you are ready to move if the dog could repeat this.  It wouldn't hurt to talk to the landlord.  If the flooring needs to be replaced, they might prefer something different, in which case they might agree to just taking an amount you would have spent on the repair (without you having to be responsible for the work).