Assuming it's a 3 man fireteam and budgeting isn't really an issue
> navigation tools in conjunction with GPS
>at least one pair of Nods
Pointman will be equipped with a semi auto shotgun with hard lead or tungsten slugs. If nothing goes terribly wrong, he'll be the one making the kill. The other two will have AR platform .308s and will support the pointman by watching the flanks and rear. All with carry handguns in 10mm.
We'll be treating this like a Tiger hunt, unless some real spoopy shit is specified. First few days will be focused on finding the beast's lair. If it's an animal, it sleeps somewhere. Using the axes, a palisade will be set up around the camp to give a small amount of security and tripwires set up for the hope of an early warning. Shifts will be taken to keep watch. Chances are though, whatever it is will be some form of ambush predator, which will naturally wish to refrain from attacking an armed group of (presumably) competent aggressors. If it has human like intelligence, it will understand that it's best bet is to monitor and set traps, rather than attack. During the search the group should refrain from leaving each others sights and maintain radio contact for when it's unavoidable.
If the Animal's lair is subterranean, Use pipe bombs to collapse it's tunnel If not, use pipe bombs to herd it into pointman's kill zone. The problem with these things isn't killing them, it's finding them.
If it's something altogether magical, we'll just burn the forest down and call it a day.