Is it permissible to pray two rak’ah of sunnah at the masjid even after the iqamah for Fajr has been pronounced; after which they join Fajr prayer with the imam?

That is not correct, because we are allowed to pray those sunnah after the fardh. The recommended two units of Sunnah prayer before the Fajr prayer can be done after the fardh, though normally Prophet Muhammad (ﷺ) said:

Narrated by Imam Muslim in his Saheeh (832) from ‘Amr ibn ‘Abasah, that the Prophet (ﷺ) said to him:

"Pray Fajr, then refrain from praying until the sun has risen and become high, for when it rises, it rises between the horns of the Shaytaan and at that time the kuffaar prostrate to it. Then pray, for the prayer is witnessed and attended until the shadow of a spear falls directly north (i.e., noon). Then refrain from praying, for at that time Hell is stoked up. Then when the shadow moves forward, pray, for the prayer is witnessed and attended, until you have prayed ‘Asr. Then refrain from praying until the sun has set, for it sets between the horns of the Shaytaan and at that time the kuffaar prostrate to it."

This was done to prevent people from misunderstanding as; it may appear to others as sun worship because those who worship sun offer prayers at those times. However, it’s recorded authentically that:

Umm Salamah, narrated that the Prophet (ﷺ) prayed two rak’ahs after ‘Asr. He was asked about them and he said: "O daughter of Abu Umayyah, you asked about the two rak’ahs after ‘Asr. That is because some people from ‘Abd al-Qays came to me and distracted me from the two rak'ahs that come after Zuhr; these are they (i.e., I prayed them just now)." [1]

So praying the sunnah prayer of Fajr after Fajr prayer, is permissible.

Moreover, Ibn Maajah (1154) narrated that Qays ibn ‘Amr said: the Prophet (ﷺ) saw a man praying two rak'ahs after he had prayed Fajr prayer. The Prophet (ﷺ) said, "Are you praying Fajr twice?" The man said to him, "I did not pray the two rak’ahs that come before it, so I prayed them now." The Prophet (ﷺ) remained silent.

So as we have an option to pray after Fajr, we should not have gone to other options for performing these Sunnah prayer as to the point to miss the part of our obligatory prayer.

Reference:

[1] Narrated by al-Bukhaari: 1233 and Muslim: 834