Praise be to Allah.
Your going to the mosque and attending Taraweeh with the congregation and meeting your Muslim sisters are all good things, praise be to Allaah. The fact that you are helping your brother to do this as well is another good deed to be added to the others.
There is no conflict between this and your praying tahajjud at the end of the night. You can combine all these good deeds, by doing one of the two following things:
1 – Pray Witr with the imam, then when you can do tahajjud after that, prayer whatever Allaah wills for you, two by two rak’ahs, without repeating Witr, because one cannot pray two Witrs in one night.
2 – Delay Witr until the end of the night. When the imam says the salaam at the end of Witr, do not say the salaam with him, rather stand up and do an extra rak’ah, so that your Witr will then be at the end of the night.
Shaykh Ibn Baaz (may Allaah have mercy on him) was asked: When some people pray Witr with the imam and the imam says the salaam, they stand up and do another rak’ah so that their Witr will be at the end of the night. What is the ruling on doing this? Is it regarded as finishing the prayer with the imam?
We do not know of there being anything wrong with this. The scholars stated that there is nothing wrong with doing that so that one will pray Witr at the end of the night. It is still true that he has stood with the imam until he finishes, because he stayed with him until the imam finished, then he prayed an extra rak’ah for a legitimate shar’i purpose, so that his Witr will be at the end of the night. There is nothing wrong with that, and that does not mean that he did not stay with the imam, rather he stayed with the imam until he finished, but then he did not finish with him, rather he delayed a little. End quote.
Majmoo’ Fataawa Ibn Baaz, 11/312
Shaykh Ibn Jibreen (may Allaah preserve him) was asked a similar question and he replied:
It is better for the person who is praying behind the imam to follow the imam until he finishes Taraweeh and Witr, so that he will have stayed with the imam until he finishes, and the reward for spending the whole night in prayer will be recorded for him. This is what Imam Ahmad and other scholars did.
Based on this, if a person prays Witr with him (the imam) and finishes with him, there is no need to do another Witr at the end of the night. If he wakes up at the end of the night he may offer whatever prayers Allaah has decreed for him, two rak'ahs at a time. And he should not repeat Witr, because there should not be two Witrs in one night.
Some scholars regarded it as preferable to pray Witr with the imam and do an additional rak’ah, by standing up after the imam says the salaam and then doing another rak'ah and saying the salaam, and then praying Witr at the end of tahajjud, because the Prophet (peace and blessings of Allaah be upon him) said: “If any one of you fears that dawn is about to come, let him pray one rak’ah as Witr …” And he said: “Make the last of your prayer at night Witr.” End quote.
From Fataawa Ramadaan, p. 826
The Standing Committee also issued a fatwa saying that the second matter is good.
Fataawa al-Lajnah al-Daa’imah, 7/207
We ask Allaah to grant you guidance and steadfastness.
And Allaah knows best.