A base ball match will come off to-day on Gamble's Lawn, at one-half past 2, between the Morning Stars, jr., and the Empire, jr., base ball clubs. Thursday, September 11, 1862.-Missouri Republican, September 11, 1862
I've found evidence that the Morning Stars were still active in 1861 and this seems to suggest that they may have been active in 1862 as well. If the Morning Stars, jr., was affiliated with the Morning Star Base Ball Club, which doesn't necessarily have to be true, then this is evidence that the club was active into 1862.
Also, I think it's worth mentioning that Gamble Lawn appears to have been the most popular site for games during the war. I'd have to go back through my notes to confirm this but I believe that more games were played there during the war than at Lafayette Park or the Fairgrounds.