BNP has always been a secular party irrespective of how they are portrayed in Indian media. Its ranks are filled with retired army officers and businessmen. Even Ziaur Rahman wasn't an "Islamist" in the conventional meaning of the word. BNP party charter does not have requirement to follow Islamic principles in their political activities, something Jamaat and other Islamic parties in BD explicitly have in their party constitution. If anything, one can accuse BNP of being an elitist party, as their grassroot level organization had always been weaker compared to BAL and Jamaat.
The reason they are termed "Islamists" is because the uber liberal leftists in Bangladesh hated them for their alliance with Jamaat for 2001-2006 rule and out of their unconditional love for BAL. BAL used that term to portray BNP negatively in the International arena at a time when the global war on terror was on full swing. We also saw the rise of JMB during that period, conveniently timed for BAL. And all this received the complete blessing of Indian govt. and media.
I remember watching Veena Sikri (former ambassador to BD) immediately after the initial protests started in July, coming on television and blaming the "Islamist" BNP-Jamaat for the instability. This again demonstrates the Indian govt failure to stay neutral and support Bangladesh as a neighbouring state instead of supporting a specific party in Bangladesh i.e. BAL.
Sorry I digressed a bit there.
That alliance with Jamaat is no longer in place. BNP now is as much of an "Islamic" party as BAL is.
Jamaat on the other hand has gone through a rennaissance under their current leadership. Their current Amir Dr Shafiqur Rahman (physician) is probably the most eloquent and qualified leader of a major political party BD has seen since Ziaur Rahman. He has been actively meeting with leaders of Hindu community and instructed his party men to guard temples and Hindu households, following the deterioration of law and order after Hasina's fall. This is something previous Jamaat leadership hadn't done. He went as far as to say that we should get rid of the majority Muslim minority Hindu mindset and consider all as Bangladeshis. I view this as a very positive change in the political landscape of Bangladesh. You need leaders with positivity and forward thinking, instead of those who are stuck in the past with petty vengeful mindset like Hasina.
The following videos are in Bengali, if you have time Dada please watch both. I found it a very refreshing change to hear a BD politician speak like this and connecting with the people with opposing views -