There are ways to get the manufacturer to reach more, it is a matter of finding the right party and setting up good communication lines, preferably directly with engineering. Side sliding cams and cryogenic deburring are definitely a great option to look into as well. And please understand, if you angle the surfaces of the mold core, since the part shrinks away from the cavity, it will still release from the core even at low or zero draft for some geometries. It is a matter of the risk and time you want to invest to reach these solutions. I have had a part with 0.25 draft over 3 inches approved for injection molding - does it look exactly straight, no, but still very good and the general eye does not pick up that difference especially if it is well surfaced.