So, is it possible for me to extend my Probation Period on request?
Anything is possible, although I have never heard of anyone asking to stay on probation - never mind being granted such a request.
Probationary periods are intended to be the time where both sides get to know each other, and decide if there is a good fit.
At the end of the period, deciding that it isn't a good fit is easy, and departure is quick.
If the "benefit" you are seeking by staying on the probationary period is the ability to depart quickly, it seems unlikely that a company wanting a permanent employee would be willing to extend. Why should they bother when you are giving signals that you won't be around very long? The better move from the company point of view is to part ways and quickly bring in another candidate to start their probationary period in hopes that she/he will stick around.
I usually don't offer probationary periods when I hire. The few times I have, I wouldn't have extended the worker if they asked for a probation period extension.
What would be the most polite way to do so?
As with most things, the best choice is to communicate with the people that matter, and are in a position to act on what you want.
Go to your manager and politely ask about the possibility of extending your probation. Be prepared with an explanation when you are inevitably asked "Why would you want to do that?" Be prepared if the answer is "No."
An alternative would be for you to ask for a different status.
You could offer to remain as a "temporary" employee, perhaps until a new permanent candidate is found. That way, you could retain your easy-out "benefit", but still provide value to the company.
It doesn't sound as if you really want to be on Probation any longer. You just want one (or some) of the things that go along with being on Probation. Talk to your manager, and be clear about what you really want. It seldom hurts to ask.