Could you give more information?
Why was there 5 homes in 5 years? Was there similar behavior?
Does it relate to food or feeding?
It seems obvious that the new rescue beagle does not consider itself as submissive to the established dog in the pack order. It may be she will end up the dominant and things will settle down but if your current dog doesn't want to accept this lower position you can have some potentially bad fights.
I do not know why there were multiple homes. My stepson adopted her and had her for 3 months, and it was too much with just buying a new home and having another dog. So I wanted to give her a permanant home. I know they owners before him only had her 2 months because he had cancer and I dont think they could deal with her bark, I believe they lived in a apartment. According to my stepson, there was no similar behavior with her and his other dog. It absolutely has to deal with food, so we are feeding them in separate rooms, but as soon as she shes him, even without food around, she goes after him, and it is only at night, all day they get a long fine playing.The other dog is a 14 year old Benji type, wouldnt hurt a fly.
I'm not sure how long the food protection instinct can linger but I'm thinking try to feed at a different time (maybe in the mornig but still separate) or even skip a day of feeding to see if the behavior still occurs. It doesn't hurt a dog to miss one day of feeding and it may clarify the problem.