Couldn't they try loaning some of that army as mercenaries to some foreign power?
Errr... possibly? After the wars ended they'll be no shortage of unemployed samurai and ashigaru. Especially for those who were fighting on the losing side. If its a choice between having the ex-soldiers live as unemployed vagrants, banditry or going overseas in search of work, many might consider the latter option. Especially if its actually legal to leave Japan. The Siege of Osaka (1615) managed to attract what was apparently a sizable amount of mercenary soldiers to the rebel forces, simply because a lot of them didnt have anything else to do anymore after 15 years of peace and budget cuts.
Even as late as 1650 (a good half century after unification of Japan and the Battle of Sekigahara and 20 years after the last major military conflict involving Christians) the Tokugawa Shogunate was still trying to figure out how to deal with all the unemployed samurai wandering around. Though it that case it was as much because successive Shoguns ended up being a bit too happy with the revoke title feature of their offices.
It depends a lot on why anyone needs Japanese mercenaries though. The Tokugawa policy of isolation was as much because of fear of christian influence as it was about avoiding a repeat of the 6 year Korean War. The threat of foreign espionage, combined with an idea that foreign wars were a waste of time (over 100 thousand Japanese were left in Korea or the waters around it. In a war they didn't even win), plus protestant Netherlands talking smack about Catholic nations, led to a specific mindset that decided to cut off all political ties and keep Japan's involvement in the outside world limited to economics. It's not that hard to butterfly away.
And as said, there was a 15-30 year gap between demobilization (standing armies are expensive) and the decision to close the ports otl. So I wouldn't be surprised if in that time a few Japanese mercenaries didn't find themselves working for South East Asian kingdoms or European companies. Tangibly related, on the other side of the cultural exchange Nobunaga had an African slave who became his bodyguard (and a black samurai). Also guns. The daimyo loved guns.
Back to the op tho, there's so many places where a Uesugi Shogunate can go differently way before that. Uesugi was supposiedly a devoted Buddhist like Delvestius says so would be less friendly to Christians than Nobunaga. And then there's the (imminent) succession crisis to deal with.