The Small House at Allington concerns the Dale family, who live in the "Small House", a dower house intended for the widowed mother (Dowager) of the owner of the estate. The landowner, in this instance, is the bachelor Squire of Allington, Christopher Dale. Dale's mother having died, he has allocated the Small House, rent free, to his widowed sister-in-law and her daughters Isabella ("Bell") and Lilian ("Lily").
Lily has for a long time been secretly loved by John Eames, a junior clerk at the Income Tax Office, while Bell is in love with the local doctor, James Crofts. The handsome and personable, somewhat mercenary Adolphus Crosbie is introduced into the circle by the squire's nephew, Bernard Dale. Adolphus rashly proposes marriage to portionless Lily, who accepts him, to the dismay of John Eames. Crosbie soon jilts her in favour of Lady Alexandrina de Courcy, whose family is in a position to further his career. Lily meets her misfortune with patience, and remains single, continuing to reject Eames, though retaining his faithful friendship. Bell marries Dr Crofts, after refusing an offer of marriage from her cousin Bernard.
As with all of Trollope's novels, this one contains many sub-plots and numerous minor characters. Plantagenet Palliser (of the "Palliser" series) makes his first appearance, as he contemplates a dalliance with Griselda Grantly, the now-married Lady Dumbello, daughter of the Archdeacon introduced earlier in the Chronicles of Barsetshire.

Question: For whom does Bell refuse an offer of marriage?
Answer: Dr. James Crofts.
Alternative answer: Bernard.