Notice how I didn't mention the knife.
What's the point of a limited ammo challenge if you can use weapons that don't abide this rule.
That said yeah, 200 shots between your Polar Star and Fireball isn't gonna be a fun time on a number of bosses.
In fact, if we assume we get 100 shots from the Polar Star and 100 from the Fireball, we're looking at a total of 1000 damage (400 from Polar, 600 from Fireball), and this is assuming you keep both at level 3.
This might be manageable early on (though you'll have to be very efficient with one boss before Fireball), but you won't be able to beat the doctor unless you can recharge between fights (and even then, things only get harder afterwards...).
I'd say that putting a refill after the doctor and giving each weapon 200 ammo would give you a proper chance with some margin for error (and this is still not considering projectiles/minions), but only for beating the game on the normal ending.
EDIT: Then again, the sanctuary includes the ability to have curly fire too but wether that fits for a challenge like this or not is up to you.