Why bother fighting the Predator?

The same reason Billy stayed. To make a last stand against this creature they knew was serious business but believed it could be and needed to be dealt with. He was tired of being hunted.

Even if he escaped he was returning to a life of depression and trauma after losing all his friends. So deep down he just wanted to die and if he won he would have that satisfaction to help him keep going.

Lust for revenge.

The Predator would have shot the chopper down and just killed everyone in it anyway.

Literally, this is what happened to Dillon's team beforehand, with the pilots skinned and strung up in the trees.

It was foreshadowing that if Dutch didn't deal with the Predator, he would have died trying to escape like Dillon's men before them.

I would say the mud would have lost its cloaking ability quite quick. Once he started moving around it would have dried out and he would be a sitting duck.

He said it early on, "We have to make a stand here, or it will pick us off one by one". Once the chopper arrived the Predator would have taken it out. He realized his only chance was to stand and fight.

Several factors: He didn't want it to survive and go on hunting other humans, he was angry and wanted revenge for the deaths of his comrades, and of course, his ego was on the line. HE was the biggest, baddest, meanest predator in that jungle, and he'd put his life on the line to prove it!

And the Predator realized it at the end, took off its armor and wanted to fight one to one. It was a contest between alpha... not males, since we don't now about the Predator's sex, so alphas.