In your case, a 301 redirect is appropriate. The 302 is temporary and does not yield the same results. A 301 is more traditional and correct.
Adding another domain name normally will not yield any benefit, however, from a marketing standpoint, using it in a marketing campaign makes sense. While as an IT professional, I cannot stand adding another domain name, the marketer in me says YES! Total win!
The only thing I would do from a cautionary standpoint would be to create a canonical link on example.com/path/to/page that points to itself. While there is generally no danger in your scenario, when a page can be found via multiple URLs, it is always wise to chose one as the original and use a canonical tag. In this case, you do not have a choice as to which URL to use.
Here is Google's page on that. There is an example under the heading: Indicate the preferred URL with the rel="canonical" link element