Buddha = Burkhan (Бурхан)
Buddhism = Burkhan Shashin (бурхан шашин). Shashin is a loan word via old Uyghur ultimately derived from the Sanskrit śāsana.
PS. I know you asked for the phonetic version only, but I included the Cyrillic because there is no universally accepted transliteration scheme for Mongolian (classical or any of the contemporary languages) so if you want to search further being able to cut and paste the cyrillic forms makes it easier to find information. You can see the above transliterated as Burγan Šašin, Burġan, and so on.
So Buryan, Burgan, Burkhan are how it is pronounced - many thanks. There may be regional dialects in Mongolia, like Tibet anyway.
Those are just three ways of transliterating the contemporary pronunciation of the same Mongol phoneme, represented in cyrillic by х. The γ is a Greek gamma, not the latin y, and it and the 'kh' and the ġ (with the diacritic ̇) are all trying to represent that same sound (in the modern Mongolic languages they are http://en.wikipedia.org/wiki/Voiceless_velar_fricative
, there is debate about how it was pronounced in earlier Mongolic languages). I just put them up to show that there is a plethora of ways to transliterate the classical script and the cyrillic script into latin based alphabets in order to explain why I put the cyrillic in.
Your inclination regarding dialects is correct, and there are several extant Mongolic languages besides the big two Khalka and Buryat.