Heather,had a look for Rose Mulhern without any success.Considering that she was born in 1800, it is highly unlikely you will find a baptism record as they are few and far between.In the National Archives in Dublin they have microfilmed parish registers but for all the places where Mulherns and McFadyens are prevalent in co Donegal the registers begin around 1870 with just a couple prior to that but no earlier than 1848.The various parishes may hold earlier records that they didn't allow to be microfilmed in the 1950s but you'd have to contact individual parishes and inquire.The parishes where McFadyen/McFadden are most prevalent are Clondavadog,Aughnish,Clondahorky,Kilmacrenan,Mevagh,and Tullaghobegly.For the Mulherns its parishes Inishkeel,Templecrone and Tullaghobegly.In Griffiths Valuation done in 1857 there is both a Rose and a John McFadden listed in the townland of Dunmore,parish of Mevagh.Also,I found 2 deaths for Rose McFadden in Dunfanaghy 1871(b 1791)and 1873(b 1804).For John McFadden 4 deaths in Dunfanaghy;1864 (b 1805),1864 (b 1809),1865 (b 1795),and 1867 (b 1802).In Dungloe R.C. cemetery there are loads of burials for McFadyens and Mulherns and in Dunfanaghy registers loads of McFadyens marrying McGinleys.It is entirely possible that Rose's family and Anthony's family either knew each other or were related.I found this and am enclosing it as a point of interest.
#44 14 Nov 1855 at East Muir.(Scotland)R.C. ceremony.Thomas Bryston, currently and usually residing in Baillieston,a collier,widower,this is his second marriage,currently has no children noted as either living or dead.Born Oct 1831 in co Donegal to parents John Bryson,ag labourer,and Catherine Bryson ms McGinley (deceased).to Ann McFadden,currently and usually residing in Baillieston.30 yrs of age,a spinster,female,born co Donegal November 1825 to parents Dennis McFadyen,a bottomer,and Margaret McFadyen ms Boyle.signed Patrick McLaughlin R.C. clergyman,Dalbeth Cottage,Dalbeth by Glasgow.witnesses Daniel Coyle and Ellen Fairy.If I was a betting man I would say there is a high possibility that Ann and Rose are probably related somehow. The only thing really to do is either write to parish priests inquiring about records or visit Donegal and see if you can locate a cemetery record or parish register.Happy Hunting,Kim