Is it permissible for me to have my teeth treated and fitted, by paying the total cost in installments to the bank with an added profit for the bank, knowing that I am in urgent need of treatment and do not possess the amount in cash?

If the intention is to borrow money from the bank with interest, then this is not permissible and there is no necessity for it. One should begin treatment with what is available, or seek a benevolent loan (qard hasan), or engage in a permissible transaction such as Tawarruq.