Well, it turns out, that after 4h of searching how to do it I had an epiphany and made exactly what I needed.
I have created windowed and fullscreen overlay that gives the ability to enable/disable a crosshair made specifically for PS1 Medal of Honor.
Here is the overlays cfg:
overlays = 3
overlay0_overlay = img0.png
overlay0_full_screen = true
overlay0_name = crshr_off
overlay0_descs = 0
overlay1_overlay = img1.png
overlay1_full_screen = true
overlay1_name = crshr_wndw
overlay1_descs = 0
overlay2_overlay = img2.png
overlay2_full_screen = true
overlay2_name = crshr_fscr
overlay2_descs = 0
By assigning a button for overlay change (Settings > Input > Hotkey Binds > Overlay next) I can switch between crosshair png image files that I’ve made.
img0 - crosshair disabled, it is an empty png file
img1 - crosshair for windowed mode
img2 - crosshair for fullscreen