Charge customers based on weight that is calculated after items have been purchased

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I have a client that is a farmer with a little store front to sell their meat and products. They want to sell their meat online based on final weight. We can easily setup a cost per pound so if a customer purchases 1 pound of chicken and each pound costs $10 the total would be $10. However, the problem is that the chicken doesn't get cut until the order is placed (farm to plate) and when the chicken is cut, their is no ability to make sure each cut is exactly 1 pound so it may be a bit larger or a bit smaller than a pound (just like at any meat shop).

We are unsure if there is a way to handle such ordering so that the customer chooses the amount of pounds they would like to purchase, but is not charged until after the chicken is cut and weighed. The customer would know approximately how much they will be charged but there will be a variance on if they pay a bit more or a bit less. I would imagine that the variance would have to have a min/max type of deal and the customer would have to be notified and have to agree to the final price prior to processing of payment.

Would we just set up manual payments and process the payments after the final weight is in or is there an easier way to go about this?

I appreciate any advice that will lead us in the right direction on this one. Kind of a doozy to us.