@Reimu Hakurei here are my thoughts on a possible compromise:
When I hear the phrase "forge god", I think about figures like Hephaestus, Goibniu, and the Norse Dwarves. Legendary smiths whose main narrative purpose is to equip the heroes/protagonists with powerful godly weapons or enchanted artifacts, and who are heavily associated with "metalworking" in its various forms (with a particular emphasis on tools of war, which could be an interesting motivation for the primordial God of War — killing off the deity who was enabling his/her foes to fight back against superior numbers and savagery). Since you say you'll be making "god-forged items" for people, am I correct in assuming that this is somewhat in line with your own ideas?
My own domains of "Invention" and "Advancement" would be more along the lines of developing technologies to improve the condition of the masses. Inspiring the common people to look at the problems around them and try to look for solutions. Things like medicine, infrastructure, transport, communications, energy, education, political systems. In other words, the common dream of mortals everywhere to improve their lot in life, and make a lasting contribution to civilization.
Does that sound like something that could work for you?