[PureHarvest]

One vote for mowing it as low as you can and leaving the clippings on top.
The amount of biomass you are gonna get from mixing the clippings in is not worth ripping up the web of life that is building under that soil. Not to mention that as soon as you do this, air is introduced into the soil and oxidizes the humus/organic matter you already have.
But do not fear! The massive root systems that your rye produced are already in your soil and will add plenty of carbon and organic matter to the soil right where they will decompose after you mow (kill) your rye.

There is a big caveat. You may have a challenge down the road when you try to plant into the brown stubble and rootmass below.
It takes time for this stuff to decompose. Don't let the area dry out if it stops raining.
Now would also be a good time to spread your compost on top which will give a shot of microbes to the area to break things down.
If you can get an inch or two down that should completely smother the area and in a few weeks you should be good to plant. The compost will mix in the soil when you dig your planting holes. When all your plants are in, mulch everthing with straw or brown leaves.