L Vanice nailed it.
Pretty much all steels have similar Young's modulus, and work-hardening, or hardening by heat treat cannot significantly increase the Young's modulus.
Strength and hardness do not equal stiffness.
Age-hardening can be obtained using special alloys, but 304 isn't one of those alloys. 300 series (austenitic) stainless steels cannot be hardened by heat treatment. They can be hardened by cold-working. But, as L Vanice pointed out, hardening doesn't change the Young's modulus. There is no way to make your 304 part stiffer using any treatment or process. You MUST use either a thicker 304 (I think Red James' suggestion is a good one), or use your current 304 sheet, but double it up to give you twice the thickness of metal*, or a different steel. Alternately, you might put ridges in the part to act as gussets. Or you could make the part wider.
Cro-Moly steels are a bit stiffer than chromium-based steels, but only by about 10%, and they don't have 304's chemical/corrosion resistance.
In sum, you cannot process this problem away using current design. You must change the material thickness, or change the design (thicker sheet, or doubling up the current sheet, or make the part wider, or add ridges/gussets to the part, etc).
*Note that Red James' suggestion, to use thicker sheet, is a better and stiffer solution to doubling up the 304.