All perfect praise be to Allah The Lord of the Worlds and may His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
In principle, a proxy should act in the best interest of the person who had vested that power in him. The Fuqaha` have agreed that the proxy is an entrusted person. Sheikh Al-Islam Zakaria Al-Ansari(May Allah have mercy on him) said, "The Wali(Islamic legal guardian) or the Wakil(Proxy) should do only that which is beneficial to the person who had deputized him."{Asnaa Al-Matalib, V.2/P.46}.
It is prohibited for the proxy to collect a commission against reducing the rent since this is an act of treachery. In this regard, Allah, the Almighty, said, "O ye that believe! betray not the trust of God and the Apostle, nor misappropriate knowingly things entrusted to you."{Al-Anfaal, 27}. It is also prohibited for the tenant to pay that commission because doing so facilitates eating up people`s wealth unjustly. Almighty Allah said, "O ye who believe! Eat not up your property among yourselves in vanities"{An-Nisa`, 29}.
In conclusion, It is prohibited for the tenant to pay a commission to the owner`s proxy against cutting down on the new rent because doing so facilitates treachery and eating up people`s wealth unjustly. And Allah knows best.