I too agree that Arjun could also go with her but with him around she won't focus on anything else. She'll only be concerned about him his food his comfort and only him. Besides he needs to be in Heritage house to make sure that Mansen Khushal and Yuvraj dont ruin Sugandhi creations.
Lastly I think bhoomika has a complex about education she doesnt think she can do it and because of that she always accepts peoples taunts. Yesterday she herself said to arjun Ill learn whatever u want me to but when I dont know this reading and education is not my cup of tea. This clearly shows she has an inferiority complex and doesnt think she is capable. She needs to do this for herself and Arjun. He loves her and will never look down on her but she cant afford to keep making these silly mistakes and she cant rely on arjun 24/7. He will not be able to be with her all the time and she cant afford to get pressurised by Mansen when Arjun isnt around. So I think this lil bit of separation of Arjun and Bhoomi will be good for them. Plus I cant wait to see Bhoomi in a different form as an educated lady. Bhoomikas character will undergo a dramatic transformation... maybe not externally but internally for sure.