My solution to the bedding issue would be a whole new bedding production chain.
CLOTH BEDDING -
Requires:
1x Any Cloth
Value: Dirt Cheap
Under my amazing new but likely already suggested or otherwise thought of bedding system, this would be the most basic of beds. It would be little more than a large, flat bag stuffed with plant fiber. Think of it as a sleeping bag in a way. It's not especially comfortable, but it's just fine for your average dwarf.
STONE BED -
Requires:
1x Any Cloth
1x Any Stone
Value: Low
Slightly better than just bedding on the ground only in that it gets your sleeping dwarves up off of the ground. As you can see, there's still cloth used. That's because it's basically an elevated stone-slab frame with bedding laid on it. Nothing special, though if it were made of an attractive variety of stone it might look nice in the room of one of your dwarves.
WOODEN BED -
Requires:
1x Any Cloth
1x Any Wood
Value:
Moderate
Another step up from stone, we find wood. A wood-frame bed understandably has a bit more give to it than a stone slab or sheet rock, and thus offers greater comfort. A four-post wooden frame also presumably looks a bit better than a slightly elevated platform made out of rock, meaning that the bed itself is more attractive, warranting a higher value in trade along with the comfort factor.
METAL BED -
Requires:
1x Any Cloth
1x Any Metal
1x Any Fuel (Unless produced at a magma forge)
Value:
Low to Extravagant
Last we have the trusty metal-frame bed. This would offer a lot of the perks of a wooden bed in addition to being very pretty if made out of a valuable metal. Alternatively, you could just start churning out bronze beds to provide your fortress moderate-quality bedding without having to cut trees or import wood. I'd imagine things like golden or platinum beds would be reserved for nobles and total snobs though.
This system makes it possible for bedding to be produced from pretty much anything provided that you have cloth to make the bedding from and a material to stick it on. It makes trees a bit less valuable since you no longer actually need wood, but if wooden beds were thought of more fondly than crude common-metal beds and stone beds, there'd still be an incentive to use them in order to boost the value of a bedroom. (Or if you built onto an aquifer and are at a loss for stone.)
No one building material should be deemed one hundred percent necessary in this game, really. Just because we might prefer to use a certain building material doesn't mean we should have to - or always be able to. That's just how things work. If I'm stuck in some horrible place where trees can't grow and all I have is dirt, rocks, and plants I can thresh fiber from, I'm going to find a way to make a bed work there. That's just how things work. I don't think the game would suffer at all from abundant bedding, personally.