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There are two great questions here: one regarding analytical rigor and one regarding fair compensation. I will address each in turn. Regarding analytical rigor, I urge you to distinguish between analysis and presentation. Policy actors seek out scholars because of the rigor we bring to our analyses. Yet they do tend to prefer reading reports that are briefer, clearer, and more direct than academic articles. The key for scholars, therefore, is to conduct the same rigorous analyses we do in our scholarly pursuits but to present the results in ways that reach policy actors. To do this, I suggest the following. First, be clear on what the organization expects and on what you can deliver. Consider asking if the organization can share any unclassified reports that they previously commissioned to get an accurate sense of what they expect. Be equally transparent about what you can produce and make sure that the two align. Second, consider producing two deliverables: first, an easily digestible piece that will be the organization’s primary reference, and second, an appendix or separate document that includes the full analysis and can be referenced by readers who have clarifying questions or seek additional details. Regarding compensation, first, do be clear about what your expertise does and does not include and allow the organization to judge whether you are the right scholar for the job. Second, recognize that what you consider to be “watered down” is unlikely to be so in the organization’s eye. If you and the organization are clear about expectations on both sides, then I suggest you trust the organization to compensate you in a fair manner. Of course, if you feel that the fee under compensates for the job commissioned you should consider raising your concerns to the organization.

If the project is interesting to you and you think you can make a contribution (and it sounds like that is what you think) *and* you have the time, it seems to me that you should accept the contract. More specific replies to your questions are below:
1. It's quite possible that the reason you were asked to consult is precisely because of the broader perspective you bring to the project. If you have reservations on this score, you could certainly reach back to the people who invited you and be clear about the extent and limits of your expertise.
2. You should conduct a robust and rigorous analysis as you are trained to do. But it is important to communicate your analysis and results clearly, without jargon, and in a way that will be legible to your interlocutors in the international organization. Here, having a very clear executive summary is key. It is possible that your contribution will be watered down, but think about the counterfactual - would they be missing more by not benefiting from your contribution at all?
3. You should be compensated for your time. It is typical for scholars to be paid by international organizations for this kind of work, which sounds extensive. I would not have ethical qualms on this question.

If you have been invited to engage in a political economy/ institutional analysis then I believe you are being asked to do exactly what you intend--a rigorous analytical approach. I think it is important for you to clarify whether that is indeed what they want. Make sure you both have the same understanding of what "rigorous analysis" means. If both sides understand the job the same way, then I don't think you have a problem. I also have to wonder whether there is necessarily a tradeoff between rigor and relevance. I should think the analytical work would inform recommendations regarding state capacity and public service delivery. Making those recommendations shouldn't lead you to water down the analysis, though you may need to frame them in a way that is understandable to your audience. If you do the work then I don't see any ethical problem with getting paid.

That said, where I would have second thoughts is in your lack of understanding of the country involved. That is the root of the ethical issues involved in your question. This may be why you are worried about the relevance of the analysis, and the level of compensation--without country expertise you may wonder if you are the right person for the job. I know that I would worry about whether my analysis fits the context, or that I might miss important factors or misinterpret some results.

The most important step to address the country expertise question is to communicate with your employers. I would guess they turned to you for this job because of similar work you have done in other countries. I assume they know what they are getting by hiring you--but you can talk to them to make sure. You could also build on the expertise you do have by suggesting that you include a comparative element in your work. You could ask them to hire a co-author with relevant country expertise, or you yourself could hire a sub-contractor to review your work. They may discover they have hired someone else already to do this, or that they have someone on staff with expertise who will be involved.

On the one hand, I think you raise sincere concerns that reflect well on you. But I also suspect you may be raising issues that the people inviting you to do this work have already considered. You may be underestimating them, and you may be underestimating the value of your own expertise.