Eames by and large is going to be relying on his illusions to alter the field to make it as disorienting as faely possible, basically weaving the arena to confuse them and maybe spook anyone who's particularly susceptible to that. Also to obscure himself from them. If he can manage to get his hands on the... ridiculous items, he'll try and improv some weapons. Breaking the chair to get what he can or making a lasso from the rope or... I dunno, throwing the ball at their face? Maybe he can use it to trip someone up who knows.

Anyone human sized (read: not the bears) he could try to use his telekinesis on to keep them away or move them where he wants them, but lbr he's not going to bother in the first match at least because who's gonna bother with a rat. It's a rat. (Regardless of who wins, that is a choice he will almost certainly regret.)

Generally his approach can be summed up as: wear them out and distress them enough to make them an easier target.