Hagaon Harav Yosef Avraham Heller Shlita, the Rosh Kollel of Crown Heights, Brooklyn and former member of the Beis Din there, wrote a essay explaining the Halachic justification for davening after Chatzos, published in “Kobetz Beis Chayenu” 11 Nissan 5760 pg. 28. The crunch of the explanation is as follows:
The Gemora (Brochos 26a) states that, “He may go on praying [Shachris] the whole day. But up to midday he is given the reward of saying the Tefillah in its proper time; thereafter he is given the reward of saying Tefillah, but not of saying Tefillah in its proper time.” Although the mainstream view in Rishonim is that of the Rashba that after Chatzos is only considered tashlumim if he accidently missed the time, the Perisha in OC Siman 89 cites an alternative view that even after Chatzos is still considered the time of davening Shachris but only that one would not receive the reward for davening on time. Rabbi Heller continues to back up this view based on many Rishonim that maintain that the time for Shachris is the entire day. Although this Prisha argues with the Mechaber and Rama, Rabbi Heller suggest that since it concurs with the view of many Rishonim, Chassidim relied on his ruling in order to have adequate kavana in their davening.
There are also several letters of the Lubavitcher Rebbe on the topic collected in “Sharey Halacha U’Minhag” OC Chelek 1 pg. 111. The Rebbe brings the halacha (OC 98:2 and Rambam Hilchos Tefila 4:15-16) that proper kavanah during davening is integral to the davening, and without it the teffila is nullified. He asserts that therefore proper preparation for davening takes precedence over being exact in the time of davening.
However, it must be stressed that all the above is only when one is involved in davening or it’s preparation from before zman tefila. There is no justification to begin davening after chatzos.
In terms of an opinion that as long as one begins before zman tefilla they may continue even after, I heard that this comes from Tosfos in Brochos 7a “Sh’ilmaley”. The Gemora relates that Biliam tried to find the exact second Hashem was angry to curse the Jews, and Tosfos asks what he could have said in that short moment. In their second answer Tosfos says that as long as he would have started at a time of Divine anger the curse would work even after. Similarly, the “zman tefilla” follows when one begins even if he finishes after.
Lubavitcher Chassidim learn chassidus for at least one hour and do hisbonenus (“meditation”) on gadlus Haboreh using the concepts explained in Chassidus. In general they are makpid to finish Shacharis before chatzos but few individuals who are real “Oivdim” may use this Heter with the guidance of a Mashpia.
This is the Teshuva of Hagaon Harav Heller Shlita:
http://www.hebrewbooks.org/pdfpager.aspx?req=48573&pgnum=36