I too was on the same road you are my friend. Unfortunately as per my research on the net, its Impossible. There was bug in the past but it has been fixed now.
In short, Instagram Private Account is secure as bank internet access.
Or, Instagram actual security for Private account is secure as titanium.
Anyways, best way you can get in, is buy building a fake account of interest and pray that he/she accepts the follow request.
Till then, I will save you time, it's impossible to get access via A private account unless the user accepts the follow request.