Why would it be beneficial for cortisol and corticosterone to exhibit pronounced anti-inflammatory activity?
Interesting question. It's easy to imagine why you would want to suppress inflammation in a fight-or-flight situation. Inflammation can cause painful swelling, fever, and it can dull mental acuity, all of which might hurt your chances of a win or escape.

After a little PubMed search, it looks like sympathetic adrenal responses (like cortisol) can actually induce a net hypercoagulability of the blood (1). So, even though inflammation does induce coagulation, it looks like the sympathetic anti-inflammatory response is not necessarily synonymous with anti-coagulation. This also makes sense in the context of Cushing's Syndrome, where hypercoagulation is associate with elevate cortisol levels, among other factors (2).