----- I am not an English teacher -----
I am not sure your suggestion is correct, because the pronoun 'that' is there.
If I understand well, the stuff should be omitted, not substituted by a pronoun.
We got a mortgage from the cheapest [house] lender.
After that, we made a [rent] offer on the house.
I am not sure I clearly understood the meaning of the sentence. were you trying to rent or buy a house?
What does exactly mean a 'mortgage' in this context?