Praise be to Allah.
There is nothing wrong with what you have mentioned about connecting intimacy to the time of Fajr, before or after the prayer, to help your wife to offer this prayer on time. This is a good intention. But you should pay attention to the rights that your wife has to intimacy and satisfaction, because waking her up for that may make her less keen, and may not be fulfilling her rights. So you must pay attention to that in that case.
The husband has a great deal of responsibility towards his wife, to motivate her to do good and help her to avoid evil, and to protect her from things that may lead to doom and perdition. Allah, may He be exalted, says (interpretation of the meaning):
“O you who have believed, protect yourselves and your families from a Fire whose fuel is people and stones, over which are [appointed] angels, harsh and severe; they do not disobey Allah in what He commands them but do what they are commanded”
It was narrated from Ibn ‘Umar (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) said: “Each of you is a shepherd and each of you is responsible for his flock. ... A man is the shepherd of his household and is responsible.” Narrated by al-Bukhaari (7138) and Muslim (1829).
Al-Bukhaari (7151) and Muslim (142) narrated that Ma‘qil ibn Yasaar al-Muzani (may Allah be pleased with him) said: I heard the Messenger of Allah (blessings and peace of Allah be upon him) say: “There is no person whom Allah puts in charge of others, and when he dies he has been insincere to his subjects, but Allah will forbid Paradise to him.”
Prayer is the greatest of the pillars of Islam after the twin declaration of faith, and those who are heedless concerning it are in grave danger, as Allah, may He be exalted, says (interpretation of the meaning):
“But there came after them successors who neglected prayer and pursued desires; so they are going to meet evil”
If your wife sleeps and misses Fajr prayer sometimes, then what you must do is wake her up, and use various means of doing that, such as explaining the ruling on being heedless about prayer, encouraging her to pray, explaining the necessity of the husband and wife cooperating in good, and avoiding things that incur sin.
The Prophet (blessings and peace of Allah be upon him) praised the one who wakes his wife up to pray qiyaam al-layl by sprinkling water on her face, so how about waking her up for an obligatory prayer?
Ahmad (7404), Abu Dawood (1308), an-Nasaa’i (1610) and Ibn Maajah (1336) narrated that Abu Hurayrah (may Allah be pleased with him) said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “May Allah have mercy on a man who gets up at night and prays, and wakes his wife, and she prays, and if she refuses he sprinkles water in her face. And may Allah have mercy on a woman who gets up at night and prays, and wakes her husband and he prays, and if he refuses, she sprinkles water in his face.”
The hadith was classed as saheeh by al-Albaani in Saheeh Abi Dawood.
This is one of the means of waking her up.
Another means is encouraging her to sleep early, and warning her against staying up late at night.
But we advise you to treat her gently and kindly, for there is never kindness in a thing but it adorns it. Reward her and encourage her if she prays, and remind her that prayer is the key to happiness and guidance, and is one of the means of abundant provision and an easy life. If that produces results and leads to her mending her ways, then this is what you want, and this is what we hope for and wish for her. But if she persists in her shortcomings, then there is nothing wrong with resorting to firmness in rebuking her sometimes, such as shunning her for that, in a manner that could achieve good results. Strictness sometimes may be the right and wise choice, as the poet said.
The aim behind all of that is to set her straight, hence we advise you to be patient with her, and not get irritated with her. Allah, may He be exalted, says (interpretation of the meaning):
“And enjoin prayer upon your family [and people] and be steadfast therein”
We ask Allah to guide us and you.
And Allah knows best.