Bell uses head circumference instead of age limits for their helmets; Nutcase Helmets does the same. This makes sense since age isn't a good indicator of helmet effectiveness (not even as adults is an user's age an accurate indication of helmet effectiveness). I doubt that in contacting any reputable manufacturer they differ from their written policies---a condition of legal liability in the US. In any case, helmet size by head circumference does make the most sense.
Using a thin cap (often made of weatherproof material) underneath the helmet should not affect helmet safety, so as long as the proper conditions for fit are maintained; it will help over all warmth and comfort tremendously (and warmth can also be a safety issue depending on how cold it is). This is common practice for a lot of helmet users out there who live in colder climes (think of skiers here, little and big alike). A big, thick, bulky hat under a helmet is a different issue entirely, and clearly not safe.
Porting a kid around in a bike trailer is already a pretty safe move, especially if it's a good trailer with a proper roll cage like a Burley, Chariot, and others. A helmet is going to be the frosting on this cake, and a good habit to get into (and in some, but not all cases, a state or municipal legal requirement). In California the law is:
http://dmv.ca.gov/pubs/vctop/d11/vc21212.htm
Youth Helmets. VC 21212
Prohibits persons under 18 from riding or being a passenger on a bicycle without wearing helmets meeting specified standards (ANSI or SNELL). Violations are punishable by a fine of not more than $25.
So make sure you have an ANSI- or SNELL-rated helmet (which you ought to anyway).
Your biggest real-world problem for a helmeted one-year old in a bike trailer is going to be comfort. Every helmet will likely push the child's head forward while riding, and pull it heavily to the side if the child falls a sleep. At one-year old even helmet-recessed trailers (like Burley or Chariot) will probably not be recessed low enough for the height of your child to be effective (anyway, they weren't for mine)---the recess pocket is about 12" from the seat bottom, and you'll need to account for slouching and slipping in the seat (the kid is probably not going to be sitting perfectly upright for very long). You can prop the child's head on the side with pillows, or find some other solution, but it will almost certainly be a ghetto/ersatz solution.