Is it legal under Sharia Law for a Muslim to take another Muslim to non-Sharia courts for remonstration?
There isn't any room for non-Shariah courts under Shariah law, as there is prohibition and severe condemnation of using any other man-made law:
وأن احكم بينهم بما أنزل الله ولا تتبع أهواءهم
And judge between them by what Allah has revealed and do not follow their inclinations
ومن لم يحكم بما أنزل الله فأولئك هم الكافرون
And whoever does not judge by what Allah has revealed - then it is those who are the disbelievers.
A non-muslim judge (for example one who is appointed over Dhimmis in some of their internal matters, according to some madhabs) does not have any jurisdiction over a Muslim:
ولن يجعل الله للكافرين على المؤمنين سبيلا
And never will Allah give the disbelievers over the believers a way [to overcome them].