There are special types of regression models for these kinds of problems.
The expected loan amount for a given individual is a function of both the probability that they respond and take out some type of loan and the loan amount they take out conditional on them responding. In mathematical terms, we can think of it as expected value of the loan amount and it can be expressed like this:
$E[loan$ $amount] = Prob(response) * E[amount | responded] + Prob(no$ $response) * E[amount | not$ $responded]$
(It seems like in your case you may be assuming that without receiving a letter an individual has a zero probability of taking out a loan, so you can drop everything after the $+$ operator, if that's the case.)
So you have to model both variables - probability and amount. The question is how. If you just go with the first approach you outlined, i.e., a basic regression for modeling counts like Poisson or Negative Binomial and include 0's in the model, then you will be assuming that the zeros and the nonzeros (positive values) come from the same data-generating process, governed by the same data distribution, and are influenced in the same manner by the same set of predictors. But models like hurdle models and zero-inflated models (e.g., zero-inflated Poisson) separately model the binary outcome of whether a count variable is zero-valued or positively-valued and the conditional distribution of positive values.
(If you want to know more hurdle and zero-inflated models, here is a couple of links to Q&A's on the Cross Validated site:
You could also manually construct those separate binomial (logistic) and count models (using separate sets of predictors if you wanted) and try to combine the results. One issue though is that you would need to model amounts only for those individuals who have previously responded (as you are looking for expected amount conditional on responding) and it would be difficult for you to calculate/estimate expected amounts for individuals who have not previously responded and for any new customers.