All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
It seems that your work falls under brokerage. Brokerage is in principle permissible with a definite wage if it does not lead to what is forbidden, like Riba (interest and/or usury), gambling and risk.
There is no difference as far as the Sharee'ah ruling is considered between dealings among Muslims or dealings between a Muslim and a non-Muslim. The Companions used to dealt with the Jews.
As regards dealing with banks that deal with Riba, then this is forbidden as this is considered as cooperating in sin, and this leads as well to consuming Riba and writing its transactions.
Allaah Says (what means): {...but do not help one another in sin and transgression.} [Quran 5:2]
Moreover, the Prophet said: "Allaah has cursed the one who consumes Ribaa (i.e. usury or interest), the one who gives it to others, the one who writes it down and the one who witnesses it." He said: "All of them are equal in sin." [Muslim]
On the other hand, it should be noted that Allaah has prohibited work in earning a living through forbidden means. Allaah is Good and accepts only that which is good. The Prophet said: “Indeed, Allaah is Tayyib (i.e. Good) and He accepts only that which is good. Allaah has ordered the believers to do that which He has ordered His Messengers to do, as Allaah Says (what means): {[Allaah said], “O messengers, eat from the good foods and work righteousness. Indeed I, of what you do, am Knowing.} [Quran 23:51] Allaah also Says (what means): {O you who have believed, eat from the good [i.e. lawful] things which We have provided for you and be grateful to Allaah if it is [indeed] Him that you worship.} [Quran 2:172] Then the Prophet mentioned (the case of) a man who has travelled a long distance – unkempt and dusty – and who raises his hands towards the heavens and supplicates: 'O Lord!' 'O Lord!', while his food is ill-gotten, his clothing is ill-gotten and he is fed with ill-gotten money; then the Prophet said: "How could his supplications be answered?"[Muslim]
The Prophet also said: "Jibreel (Gabriel) inspired in my heart that no soul will die until it would have consumed its (designated) provision and completed its appointed time (i.e. it will not die until its predestined time for its death comes), so seek your provision in moderation. If your provision is delayed, do not seek it through disobeying Allaah. What is with Allaah can only be obtained by obeying Him" [Abu Na'eem]
Therefore, dealing with clients according to a wage or a definite proportion is permissible.
You should keep in mind that you are acting on behalf of clients in dealing with the institutions, so you should advise them. But you are not allowed to add a commission to the fee determined by the bank, or take this commission from the clients or that the bank takes it from you.
It should be noted that when a Muslim woman deals with men, she is obliged to observe Hijab and not be in seclusion with them, and she must observe decency when addressing men.
We also warn against living in non-Muslim countries and that a Muslim is obliged to move to a Muslim country where he/she would be secure in himself/herself and his/her religion. The residence of a Muslim in a country whose people's manners and economics are based on what is forbidden may cause him/her to indulge in it.
He/she might die while being sinful and addicted to what is forbidden, and so, incurs Allaah's Wrath. So, it is obligatory on him/her to migrate to Allaah's spacious earth.
Allaah Says (what means): {Indeed, those whom the angels take [in death] while wronging themselves — [the angels] will say, “In what [condition] were you?” They will say, “We were oppressed in the land.” They [the angels] will say, “Was not the earth of Allaah spacious [enough] for you to emigrate therein?” For those, their refuge is Hell — and evil it is as a destination.} [Quran 4:97]
This verse was revealed by Allaah regarding those who didn't migrate from Makkah to preserve their religion. Imaam At-Tabari interpreted "while wronging themselves" to mean the people who incur Allaah's Wrath.
Allaah Knows best.