The Bulwark is an interesting ship. It's only remarkable attributes are that if you can spend the ammo it can put out a lot of damage (shield-mitigated damage; and you start with 28 missiles) and that you start with 3 rock crewpersons. . . err crewrockpersons. Rockmen are slow as Christmas but they get 150 hp and are fire immune. They make pretty good boarders.
With 28 starting missiles and a few breaks, it's easy enough to roll through the first two sectors in the Bulwark. But if things break the other way, or if you can't find anything to suppliment your combat, you could run into trouble. The Rock has a pretty strong start I think, but it's subject to mid-game problems for the reason you mentioned; if you spend too much on ammo you'll not get other, necessary, upgrades.
In my lone Bulwark win (I've only played it a handful of times) I was lucky to find a Heavy Laser Mk II early, and then found another energy weapon in the mid sectors (I think a Blaster Mk II). Those two weapons + one launcher is a very potent combo (though that's an expensive setup at 7 power). Still, it's one you can build towards gradually as the two energy weapons can take you through most of the game.
I think it's better to save money and hope for the right upgrade than to keep buying ammo. That means there will be games where you are nearly in a fail state by sectors 3-4, but that's true even if you are buying lots of ammo. And the upside of finding good complimentary energy weapons (or getting a good drone setup) outweigh the downside.