I just want to make sure I understand the situation. You have a fire-rated door at the front of the house with a small screw hole on the interior - does this mean it is an exterior door? Fire-rated doors are usually used to separate two areas of a building to slow down the spread of any fire, e.g. separate garage from living space.
If you are using the fire-rated door as an exterior door, then you are not really benefiting from the fire-rating (maybe it was chosen for the higher security that might be afforded by such a door). In this case, you don't need to worry about the repair meeting the fire-rating. I think you recognize this as you said you were "mainly looking for something cosmetic".
There are many ways to make a cosmetic repair, so the first thing is to assess what tools and skills you have. For example, do you have a grinder? If so, would you have the skill to be able to grind a screw head down smooth?
It is a change in texture that makes patches visible. For example, most painted drywall has a bit of an "orange peel" texture, but patches are quite smooth. If you want your patch to completely disappear, take care in giving your patch the same texture as the rest of the door when you prime it for painting (and please paint the whole door, not just the patch).
I don't have a new repair technique to add to the excellent suggestions above, but I would put them in the following order of difficulty - starting with easy:
- Just paint alone. Con: might not hide the hole, risk of drips...
- Put in a dap of toothpaste or drywall compound, smooth with finger, then paint. Con: Patch material might rust.
- Patch with bondo, then paint. Probably the best option to get a really good finish. (I would go with this one myself.)
- Patch with other more liquidy material such as epoxy, jb-weld, etc, then paint. Con: Would have to take door off and lay flat to apply patch material otherwise it would run.
- Install screw and grind flat. Con: a lot of work for no significant benefit. Plus would probably want to take door off for ease of repair.