I've run in to this over the years which is why I run a custom development company but we make products for other companies and one has run in to this problem. They are a SaaS company and target individuals as well as small groups. A large group that represents a lot of individuals has reached out and asked for a version of the platform that has some unique logic and workflows specific to how they manage their constituents. Conceptually these modifications can not be made to the core platform so we need to come up with a way to effectively price this arrangement. We would host and own code basically as a SaaS platform unwise to them. Primary concerns How to charge for more than just modification hours. Impact of branching the platform long term with upgrades etc. What have other smart people done in similar situations. We are reluctant to pass on the deal because of its head count and symbolic "lighthouse" name power in the industry.
Great question, I've dealt with this exact same issue. I've found the best way to look at this problem is by analyzing the problem you are solving for this "lighthouse." Is this something they would build anyway? If so, how much would it cost? Anything less than that is probably a win for them. Is this a nice to have, or a need to have for them? Are they price sensitive? How do they usually pay for this type of product/service?
If you are able to answer these questions it should help you figure out the best pricing model (up front cost + hosting, monthly service, per user/group etc..). It sounds like you have something pretty valuable that they want. I think pricing it based on "modification" is selling yourself short. If you solve a big business pain for them, they should be willing to pay you something non-trivial. My hunch modification pricing would be trivial for you.
Sounds like a great opportunity for you and your company, don't sell yourself short. You'll never get a price that you don't ask for.