Can we reconcile Ramban’s “any king not of the descendants of David is sinful” with today’s state? Would he make a distinction between king and state?

Correct, the Ramban himself in the source which you are apparently referring to (Breishit 49, 10), says explicitly that the Maccabees should not have called themselves "king", but rather "judges" or something democratic, solving the problem of transgressing Ya'akov's blessing "the scepter (!) shall not leave Judah". He details explicitly that the prohibition is to have a monarchic dynasty (son after father), which is anointed with oil and has the "aura" of kingdom, all of which are not found in the democratic State of Israel. See the Malbim's commentary on Micha 4, 8 who makes the same differentiation regarding the "first government" and the final "kingdom" of redemption. [If you read Hebrew, see my article on the topic: https://asif.co.il/download/2%20(9).pdf