sentence B below should be more efficient than the sentence A without loss of any meaning.
If we take the whole meaning to be "I lent you money," then B is more efficient and the あげる in A is redundant.
If, however, we take a more nuanced view of what A and B are saying, then A differs in adding additional information. The あげる implies the subject is doing the recipient a favor whereas the other sentence merely states the fact of lending.
Both are translated "I lent money to you"
Are they? If so, then I'd say that we're losing information in the translation of A.
B1: I lent you money.
B2: I lent money to you.
(I'd say B1 is actually more natural English than the translation you're suggesting).
A1 : I let you borrow money from me.
A2: I kindly lent you money.
A3: I volunteered to loan you money.
A4: I lent you money.
Depending on the context, A4 may be fine or it may lose something valuable and important for understanding the rest of the dialogue.