Should he propose to her even though she is not sufficiently religiously committed, but she is well mannered and obedient?

Firstly: 

You have no need to ask this girl about the details of her previous engagement or the details of her life before she graduated from university. In fact you have no right to do that at all. This is the right of Allah and it is between her and Him. You have no rights over her at this time such that you could ask her about that or demand it of her. Rather you have the right to know only what is apparent from her at present; if she appears to be good and righteous, then go ahead and propose to her. But do not ever ask her or try to find out about her. And if she does not appear to be good, then forget about her and do not ask her or try to find out about her. 

There is no one among us who does not have anything in his past, or even in his present, that he hopes Allah will forgive him for and conceal it in this world and in the Hereafter. 

Ibn ‘Abbaas (may Allah be pleased with him) said: Allaah is Forbearing, Compassionate towards the believers, and He loves concealment. Narrated by Abu Dawood, 5192. al-Albaani classed its isnaad as hasan mawqoof in Saheeh wa Da‘eef Sunan Abi Dawood 

Rather what you should do this the exact opposite of that. What you should do, if you come across a fault, is to conceal it as much as you can, and not to look for faults when everything appears to be sound, unless you are certain that there is some fauly. 

Muslim (2590) narrated from Abu Hurayrah (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) said: “Allaah does not conceal a person in this world but Allaah will conceal him on the Day of Resurrection.” 

Secondly: 

If you like this girl’s appearance and you sense that she is obedient and good-mannered and she wants to become religiously committed, and you want to marry her, then there is nothing wrong with you proposing to her and that is not contrary to the words of the Prophet (blessings and peace of Allah be upon him): “Choose the one who is religiously-committed, may your hands be rubbed with dust (i.e., may you prosper).” If a woman, in addition to being well mannered and having a good attitude, also has a sincere desire to adhere to the rulings and teachings of Islam, then she is of good character and religiously committed. Those who are of good character and religiously committed vary and the Prophet (blessings and peace of Allah be upon him) did not describe a specific level of that. Rather he said “Choose the one who is religiously-committed, may your hands be rubbed with dust (i.e., may you prosper)” after saying “Women may be married for four things: their wealth, their lineage, their beauty and their religious commitment.”

The hadeeth was narrated by al-Bukhaari (5090) and Muslim (1466) to encourage suitors to look for women who are religiously committed rather than others. 

If the girl is as you have described, then we hope that your marrying her will be good for both of you and that Allah will help you both in that. 

You could also seek the help of her brother and your relationship with him to make up what she may be lacking of knowledge and good deeds during the engagement period. 

And Allah knows best.