Alshain is capable of building Leviathan Battleships, and seems to be doing so if the Leviathan III entry in XTRO: Republic 3 is any indication. It seems the Ravens are providing assistance in the Bear's construction efforts.
Although, Alshain seems to be the only system confirmed as being able to build Warships, the Foxes in the Chainlaine Isles and the Ravens at Quatre Belle are capable of building Jumpships and maintaining Warships. So it seems that if the Sea Foxes or Ravens had the need to build more Warships they could do so relatively easily. Also, the Titan Yards of Terra could build Warships, but Stone as part of his peace initiative mothballed the Warship construction facilities, and the Republic has apparently had other priorities than reactivating them.