The premise of your question and your two demands are in antithesis to each other.
1."show that 1. they mean no harm to the country or to the people living in it" but they do, they wish to colonize them, even if they wanted to do it as peacefully as possible it won't be bloodless, even if it were, you are taking away the autonomy of a people and that by itself will harm them and even if the current leadership is well meaning there is no guarantee that the following leaders will too.
2."they want <PERSON> to become strong allies with them without replacing candidates" I believe you need to reword your question, does Vertercrank wish to ally with them or colonize them? If they are not going to interfere with the local government then they are simply allies and not a colony, this alliance can be set up with a simple meeting.
But if they wish to keep their culture in check culture will be an important factor, even if you are "benevolent" masters there will be tension, these are an independent people that have recently been defeated (assuming then didn't hand their independency away without a fight) and humiliated (being colonized). 454
How to deal with this will be very important, as we can see with real world examples people that are ok with colonization are not very open minded and tend to deal with this problem violently but this has proven to be largely ineffective.
If they want long term stability they should treat both cultures equally while creating in the colony a dependency to the colonizers, respecting their culture and bringing prosperity will on long term put them on your side against their natural adversaries (neighbors).
Edit: In answer to your edits. The best way to achieve your goals is with economic ties, your example was England and Portugal which is also a perfect example of the solution, Portugal's ties to England were mainly economic, England provided them with products they needed (wine and textiles mostly), Portugal was not willing to part with the goods and England made a lot of money from it, until Portugal was so dependent on England they could not afford to cut ties (as in the Napoleonic wars). But this would only work if the countries have good ties to each other, it could easily backfire like with England and China, where a similar situation led to the first opium war.
You mentioned using religion, that could and could not work, we have multiple religions with messianic figures and they tend to not go well together, even the same religion sometimes doesn't get along with itself.
If two entirely different but similar religions came together with their holy texts in a big council and spent years studying to come out with a new "merged" religion that could be very interesting but there are events that could very well happen in real life but we do not accept in fiction, like for example if you wrote that X character survived an assassination attempt then turned the wrong corner to accidentally meet one of the men that tried to murder him? We would not accept that in fiction but in real life it did start WWI 145