Praise be to Allah.Who has guided you to adhere to the rulings of Islam and to follow its teachings. We ask Him, may He be glorified, to divert temptation away from you, both obvious and hidden.
With regard to telling lies to the government in order to obtain a residence permit, this is not permissible, because lying is haraam. It was narrated that ‘Abdullah ibn Mas‘ood (may Allah be pleased with him) said: The Messenger of Allah (blessings and peace of Allah be upon him) said: ‘You must be truthful, for truthfulness leads to righteousness and righteousness leads to Paradise. A man will keep speaking the truth and striving to speak the truth until he will be recorded with Allah as a siddeeq (speaker of the truth). Beware of telling lies, for lying leads to immorality and immorality leads to Hellfire. A man will keep telling lies and striving to tell lies until he is recorded with Allah as a liar.” (Reported by Muslim, 4721).
The noble Messenger (blessings and peace of Allah be upon him) regarded lying is one of the characteristics of the hypocrites. It was narrated from Abu Hurayrah (may Allah be pleased with him) that the Messenger of Allah (blessings and peace of Allah be upon him) said: “The signs of the hypocrite are three: when he speaks he lies, when he makes a promise he breaks it, and when he is entrusted with something he betrays that trust.” Narrated by al-Bukhaari (33) and Muslim (55). And it was narrated from ‘Abdullah ibn ‘Amr that the Prophet (blessings and peace of Allah be upon him) said: “There are four (characteristics), whoever has them is a hypocrite, and whoever has one of the four has a characteristic of hypocrisy unless he gives it up: when he speaks, he lies; when he makes a promise he breaks it; what he makes a pledge he betrays it; and when he disputes he resorts to foul language.” Narrated by al-Bukhaari (34) and Muslim (58).
So you must always be truthful in your speech, even if that leads to you not getting the residence permit. With regard to learning the Holy Qur’an, the rulings of Islam and the Arabic language, you can do that in another country, for Allah’s earth is vast and there are many Muslim Arab countries, praise be to Allah. What is too difficult for you in one country may be easy for you in another, in sha Allah.
With regard to the need that you mention for learning the Arabic language and Islamic religion, this is something that you can do in another country. The condition you mentioned is not essential, and the country you mentioned does not have any particular advantage over any other country with regard to learning Islam or Arabic. So try hard to look for a country that will give you the opportunity to stay for longer, and where you will be able to fulfil the conditions stipulated by the government.
And Allah knows best.