I want to make an image look like it is being reflected in a mirrored, convex shape (in this case the visor of a helmet).
What sort of settings and effects would best achieve this?
Use Warp in Edit->Transform. Set it to "Bulge" and tweak the settings (you want it positive for convex). Then you can set it to "custom" it will keep your current settings but let you tweak it further by moving the handlels on the corners.
In Photoshop, you could try Filter\Distort\Spherize.