Note that the dictionary says that finesse, with the meaning you are looking for, is a noun. It is not an adjective. If it were, it could modify design. Nor is it an adverb, which could modify the adjective architectural.
So, you could say that its architecture shows finesse, or it was designed with finesse, but you can't use finesse to modify an adjective or noun.
You can use finesse as a verb, but as a verb it usually relates to trickery and guile, or cunning evasion of obstacles and mistakes. If it shows finesse, that is saying it shows style, elegance, or the characteristics of being done well. The same is true if it was done with finesse.