Edit: forgot to say thanks for suggestion! Also, here is a related article: https://hackernoon.com/im-afraid-you-re-thinking-about-aws-l...
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...
Just found the same author as OP with a clever solution here: https://read.acloud.guru/cold-starting-lambdas-2c663055589e
Having the app pre warm instances on a per-user basis is super cool -- for user-driven workloads like web servers. To make matters worse, we are serving an API that takes hits from third party streams -- so our concurrency is based on their client behavior, not something we can easily link to a session scope, like users. Tricky!
Sometimes though, you can't force a square peg in a round hole. I dislike server maintenance but docker is a decent alternative to lambdas if you can absorb the extra cost.