Good to ask this question...please do not think about throwing the chips into the creek. Not only would this disturb the ecology but this is usually against the 'law'...do you have neighbors also on this creek? Where are you living, David?
I wouldn't BUY a chipper, I would go rent one for a day. Unless you plan on getting into the chipping business? Use the chips for a mulch if you've got a 'natural' area with a large area of weeds. Great to dump on a patch of weeds and allow to decompose getting rid of weeds and then possibly a new great place for a plant bed.
As for smoking...YUMMY way to 'cook'. You need a smoker which are fairly inexpensive. Lots less than the cost of a chipper! Are your tree limbs Apple? Hickory? Alder? You would chip them twice and allow to dry in a dehydrator (I built my own that has 7 or 8 shelves each 2X2' using heat lights and a fan built into the bottom) in a single layer. Otherwise, decomposition (rot) will set in and smoking with mold is not what you want.
Barbecue is different. You throw these chips already dried onto the coals made by charcoal after the coals are red, dusted with ash and no longer in flame. Won't need to worry about barbecue if you've got a smoker, really. We've smoked our fish, made our Foh beef base smoking the vegetables with a slight char, smoked our chicken and turkey (butterfly first), started our chili peppers before dehydrating, cooked and smoked prime rib...smokers get used! Or just buy apple (my favorite wood for smoking), hickory, alder...these are the three I am familiar with...just dump your newly chipped chips in a huge pile to decompose. The bigger the pile the more heat, the more decomposition and make sure you feed your pile some nitrogen (find this stuff in the cat box section or find high nitrogen fertilizer) and a bit of moisture. Not soaking wet just some moisture now and then.
Shred with leaves on. That will actually help the decomposition. I once had the power company come dump their chips on my blackberries under my alders. 6' high piles, about 200 feet long, zone 5...those piles created this 'fog' all winter long. I threw alfalfa kitty litter pellets (nitrogen) on it as the Power Company dumped their loads. A handful or two on each 6'X6' pile?? We got lots of rain in that area so there was moisture.
In the spring, those 6' piles had become 6". No more blackberries! Nice evenly textured mulch. Very pretty. I am still in awe how quickly that happened as I know most compost sometimes takes lots longer. Don't waste this good stuff!
Tell me more about your property. Nice to have a creek! Creeks, even intermittent ones are highly regulated and have 'buffer' zones where there are laws in place of what you can and can not do even on your OWN property. For that I am glad. Riparian ecology is incredibly important and even more sensitive. Not to be used as a sewer.
In the US if a normal person catches a corporation breaking these laws they can get rich filing civil action suit. You should look into your covenants for your property. Having a creek on your property means part of your property is PUBLICLY owned. You wouldn't want your neighbors dumping anything into the water that will be running across your property. My creek was a Salmon creek for spawning. You wouldn't believe the huge salmon trying to swim up an intermittent tributary to this creek only inches deep. Even adding gravels to the shores was not allowed. But you have 'water front property'!