Q: If a shafi muqtadi reads fajr salaah behind a hanafi imam, what would be best for the shafi to do with regards to Dua e qunoot? Can he shorten the qunoot? If he does shorten the qunoot does he make sajdah sahwu? If he reads the entire qunoot and joins the imam in sajdah, does he still make sajdah?


 قال شيخ الإسلام في أسنى المطالب شرح روض الطالب: ( فإن ترك ) إمامه الحنفي ( القنوت ) في صلاة الصبح لاعتقاده عدم سنيته ( وأمكنه ) هو أن يقنت ويدركه في السجدة الأولى ( قنت ) ندبا ( وإلا تابعه وسجد للسهو ) اعتبارا باعتقاده. وله فراقه ليقنت. وقضية كلامه كأصله أنه إذا قنت لا يسجد، وهو مبني على أن العبرة باعتقاد الإمام ، والأصح أن العبرة باعتقاد المأموم ، فالأصح أنه يسجد كما لو كان إمامه شافعيا فتركه. 

From the above quotation you would learn that in terms of the rajih view in the madhhab the Shafi'i muqtadi behind a Hanafi imam in Fajr would have to make sujud al-sahw regardless of whether he managed to read the full or shortened qunut behind the imam, or not. 

The above, of course, is in terms of the rajih view. Within limits, our madhhab permits us to follow the marjuh view as well. In this case there are two marjuh views. 
The one is mentioned in the quotation above: make your own individual qunut, preferably shortened (eg اللهم اهدني فيمن هديت only, or اللهم اغفر لي), behind the imam without sujud al-sahw. 

The second is to make neither qunut nor sujud al-sahw behind the imam. In other words, follow him when he goes into sajdah in the second rak'ah, and follow him when he makes salam. This is the view of a number of eminent fuqaha of the madhhab including Imam al-Qaffal al-Marwazi, Imam al-Isnawi and Imam al-Suyuti. 
It has also been the one that I personally practice and which I advise others to practice upon.