You mention information you have >from photographs. I wonder if you have done genealogical research with those that the Mormons copied or perhaps they are not available in Sweden.
First you should find out if the Jewish records for this town are kept in Hungary or Serbia. In case they are still in the Hungarian archives in Budapest,that is a big if, then you can find the birth of the other 12 children of the Rossler/Bergenthal couple assuming they were all born in Nagy Becskerek. As women usually got married in the same town where their family lived, you should also be able to find the marriage of the sister and Erno Deim. You mention that the Deims lived in Nagymezo str in the 2nd district of Bp, but don't mention the district of their Koral str. address. I only find a Korall str in today's Budapest and that is in the 10th district close to the cemetery. If you know in what district they lived in 1918 then you can further research the Deims by district in the Budapest City archives or through childrens birth records or the couple's death records. Thus,you may find other information that may be useful, but it is a long road to
get to your great-grandfather, but it may be worth your time.
I'll not go further and I don't know what records if any are available in Serbia.