A sign like that, does not always depend on the needs of the stability or integrity of the posts or installation procedure for the final say.
First concern is a safe and secure installation, but in conjunction with an aesthetically type appearance. If you think 3" is overkill, it will look very odd with 1" for a sign that size. If you use posts too dinky, the sign will look like it has rickets. I think the 3" is fine, but I would dress up the base a little better. Anyway, I would not go any less than 2" and as mentioned, do something other than having a sh!tload of bolts going across top and bottom. If cost is not a problem, you can get post & panel kits just about anywhere these days and they have a much cleaner and better final appearance in the end.