Any way you slice it, the Death Star is much larger.
Based on the information given in the question concerning the specs of the Death Star, its radius is no smaller than 60,000 m, which gives a volume of 9.05 x 1014 m3.
Now, here is the Star Forge:
Based on this information about Hammerheads supplied by the OP,
(in particular that the length of the Hammerhead is 315 m), and based on this video also supplied by the OP,
I used some frames when the Hammerheads are between the fins of the Star Forge to give an upper bound on the diameter of the spherical part of the Star Forge as being at most 20 Hammerheads, which works out to a maximum radius of 3150 m. This gives a volume of 1.31 x 1011 m3 for the spherical part.
Now, there are six fins on the Star Forge, three on top and three on the bottom. Now, let's say that each fin has twice the volume of the sphere, which is a gross overestimate, given how thin they are. That would make the total estimated volume of the Star Forge thirteen times the volume of the central sphere, which is roughly 1.7 x 1012 m3.
This still makes the Death Star a whopping 530 times more voluminous than the Star Forge (9.05 x 1014 divided by 1.7 x 1012).
Even if I make the radius 20 Hammerheads (instead of a diameter of 20 Hammerheads) — which is another gross overestimate on the Star Forge — I still get that the Death Star is bigger, but only by about 65 times.
Another Way to Compare: If we suppose that the Star Forge were a sphere, with radius from the center to the end of the longest fin, that would be about 60 Hammerheads — but let's say 80 to overestimate. This gives a radius of 25,200 m, which is still less than half the radius of the Death Star (and makes the Death Star about 15 times the volume of this sphere), and the actual volume of the Star Forge would be much less than that of this pretend sphere.