yep, i'll chime in on this one
forklift as a compactor? nah, like everyone else said, BAD idea.
The only force is the weight of the carriage and forks and gravity, not much force to compact much.
I have a customer that does this but they use a GEHL Telehandler Boom lift with telescopic boom an attachment they had made to mount on the end of it that does the job quite well.
The boom control is a 2 way positive control (up and down) which has the capacity to force downward pressure the same as in the raise direction.
That would be your best option imo ;o)

The clue is in the name....Forklift. As has been stated lift trucks are not designed to be used as compactors because you cannot utilise the lowering capability to apply force per se.
It becomes a question of can the waste be reduced in size before it is placed in the container/skip, thereby reducing the need to try and compact it?
I am guessing that the debris is dropped in from the top and there is not the option to load through the front?

Still a bad idea. for the reasons previously stated. A forklift is not designed to do what you want it to do. Pallets are NOT designed to do compacting either. Check with you local authorized dealer for the brand of that you have and see what they have to say..

The 12' or 15' carpet rolls and construction waste sometimes rise above the edge of the container. I'll check out the skid steer loader option, but I'm not sure we have the capacity to purchase and keep it. Thank you for bringing to my attention the concern about using the forklift to compact the waste. Currently, we are just using one pallet to smooth out the on the dumpster instead of physically shifting the material on top of the container. If we limit the use of the forklift to shifting the debris which we dump with the lift into the container, it seems like it may be a good idea that additional weight by using several pallets on the forks would help keep the forks from any significant upward pressure. Based upon your comments, I'm thinking that it might be best to put the forks in the bottom rather the top of a 3 pallet assembly that can be used to shift and level out the 12' carpet rolls and construction debris on top of the dumpster. I'm looking forward to getting any other possible suggestions

Bad idea - there is no down pressure when lowering the mast assembly - gravity does the work - lowering is controlled by a flow regulator. You don't want the issues associated with - mast sections, carriage, cylinder collapsing unless you have a big check book and lots of liability/workman (women) comp insurance. Get a skid steer loader. (ask a authorized dealer if it is OK first) or something similar or hire a retired Sumo wrestler to stomp it down - if OSHA approves it. And keep a Philadelphia Lawyer on your payroll.

I'm certain there is a trash compactor around to fit your needs - they are all over for paper, cardboard recycling and even in my kitchen

I myself would not do that because when you drop down to compact the trash the chains will become loose there is a chance when pulling out the carriage might fall to ground because there is nothing holding the carriage but the chains. as for something to put on lift l don't know of anything but maybe someone can help.