I don't think your pepper sprouts will be happy outside in the cold frame. Mature pepper plants can take near freezing temps in the fall. Large seedlings (8 true leaves, 8 inches tall) can be hardened off outside if night temps are above 50. However, new seedlings aka sprouts will not grow much when night temps are near 40 even if they don't die. The soil temperature will simply be way too low to promote any kind of growth. You are better off waiting to start them than sending them out into the cold frame now. What kind of day time highs do you have?

Can you grow them indoors for a month and then set them in the cold frame?

Although tomatoes can take a slightly lower temp than peppers, I still think your cold frame is not the right place to start seedlings.

Thanks!

My plan is to keep them inside at night and on mornings with cold frame temps of upper 40s or higher (and expected sunshine) to put them out for the day in the cold frame. (The cold frame quickly heats to around 70 and then begins to autovent.)

On cool mornings (cold frame mid 40s or less) or days that look to be cool and cloudy, I'll keep them inside, up against the south facing patio door.

I'm fine if some cooler outside temps or amount of indoor light up against the patio door results in slower growth (and smaller plants to put later in the spring), I just don't want to raise plants that don't produce fruit.