Take a long branch/stick with fake leaves on it and some food/treat on it spaced out so that he would have to climb onto the stick to get them, then pull the stick out slowly and place the lil fella where you planned to keep him wile cleaning?
Have had to do similar philosophy with other animals so assume it could be a technique to try. Just tossing something up in the air suggestion wise.
while i agree this will minimize stress for the short term it will prolong the stress.
think of it this way a little stress for a long time or a lot of stress for the short term. organisms no matter how dumb have this innate ability called conditioning. though he may be hardwired (instinctively predisposed) to freak out at the sight of such a huge potential predator there are a few things you can do to ease this reaction.
mine had an eye problem when he was about 3 months old, he'd never really come out of his flex without a bit of an argument, the vet gave him eye drops which he needed 3 times a day for a week. after being handled 3 times a day for that week he seems to have learned to accept that sometimes he's just gotta come out, and normally now it's to explore which he enjoys. so he's just used to it and although he'll give out the odd hiss or make life difficult, he doesnt overly mind it. maybe try doing this with yours? a few times a day just catch him for a few minutes, then put him back? it's got to be a case of teaching them that you wont eat them
this is a perfect example. many of the hardcore keepers on here will advocate minimizing handling. and chalk it up entirely to the animals disposition if it becomes difficult. there is such a thing as being too safe.
while i agree these arent pets, theye captives or display animals being trapped in these synthetic environments they have to understand your presence is positive rather than negative. otherwise youll have some chameleons panicking at the sight of you. and with that minor heath issues compound and even being near the cage becomes detrimental to its health.
conditioning early on is integral, in my opinion, to proper husbandry.
i see these posts all the time "im having trouble handling my chameleon" "he turns dark and runs away" "he puffs up and lunges at me"
3 simple rules
move slow! be patient. never jerk away or move erratically.
approach from below, under the chin and below the base of the tail if need be, lift you hands and they will let go of the branch and grasp you. sometimes you may need to pry a foot or tail off GENTLY but this comes with practice
associate being handled with good things, going outside to wander a bush for half an hour, some feeders he normally dosnt get (hornworms are a good choice) a long shower. and most of all no scary animals or people. just peace. food, and water. and a new place to explore.
talk to it softly, this is more for you than it, it gets you in the calm mellow state of mind that is preferential for handling. especially if its being aggressive saying reassuring things like "i wont hurt you buddy, im your friend, i just want to let you outside to play or give you a treat" while they cant hear you much they should be able to sense the soft vibration of your voice. this may be calming/distracting but it works for me.
most of the time youll have em running for the door when they see you. some will be dicks forever but will get over themselves. still difficult to remove yet obviously enjoy the enrichment you provide.
give em a nice bush to hang in, set the hose to mist. sit back and crack open a beer.