F&Z - DD was also a climber, and it was an absolute nightmare for about 18 months (sorry...not what you want to hear). I used to go to other people's houses and if I took my eye off the ball I'd find her perched on top of a fish tank, or perusing their top book shelves.
I Very Seriously considered getting DD to wear a crash helmet. The huge weekly forehead bruises were getting embarrassing. But the mattress is a much better idea.
Would he entertain being held in a sling/carrier (on your back) while you get on with cooking?
Just remember that there are MANY climbers and they have falls and they are almost always alright. Even after big scary falls. He will fall at some point and you will have heart failure for a few minutes and your hair will turn grey. But he will be fine.
Also, I think the more you let him get on with it now, while he's relatively small, the sooner he'll get bored of it. He'll probably have a couple of accidents which will deter him from being too risky in future. This is somethign I didn't realise with DD...I watched her too much and she had very few accidents in the early days, so she only started having falls when she was older and able to climb higher (therefore bigger bumps).
Sorry - this is a bit of an essay. I just remember that terrible heart-in-mouth sensation and the over-burdening sense of responsibility for keeping a loony toddler safe. Good luck with it all. You'll both be FINE.