No; there's no red in the menu palette. I had the same issue with PSYCHO's sign in LPI; I wanted to use red but it isn't in the palette.
IIRC there's black (which is treated as transparent on a lot of images), three shades of brown, four shades each of green and blue, white, pink, yellow, and another shade of blue. As I said; if you use the "save palettes" option in LemMain, it'll save images containing all the allowed colors; palette_menu.bmp is the one for the menu screen. Then you can use the color picker tool in whatever graphics program you use to select colors from that.