How many instances do you want running? You could set up a separate keep alive path that sends another request to the lambda, with a variable on how deep into the keep alive request 'recursion' and break out if you deep enough. Does that make sense? Super weird and just off the top of my head.
edit: this isn't a good solution either because if you have a lambda kicking off 4 other lambdas because you want 5 running, and someone makes a request well then you still haven't warmed up that 6th and your 5 lambdas are running the keep warm code...