How should I record the value of an asset, net of a customer’s partial payment?

Your question lacks some specificity, but I’ll try to answer. As I understand it, you are both paying for an asset which, by definition, will be used to create revenue. Thus, you are effectively creating a joint venture.

Smart money would create a separate entity.  PWC created an accounting guide which discusses this concept both under US GAAP and IFRS.

Then, use the asset to create revenue and treat the entity on your books based on the percentage of ownership/control that you exercise with the new entity.