Because the space is covered on the top, I can't drop insultation down into the channel. Any rolled insultation will have to be pushed out into the channel. My contractor believes "packing" the space with R30 is the best answer as it will be fill eith insulation. (This is a red flag for me as I thought insualtion is supposed to be fluffy to trap air and work properly.) On the flip side I'm not sure how to get the insulation out into the cavity without stuffing it.

I thought I would fill the channel with R38 and then seal the channel off with a 2" piece of foam and tape. But I am unsure if R38 is needed, and if it will be possible to stuff it back there and still be loose enough to be effective.

Any suggestions?

Remember, as you stated, insulation should not be packed at any time. It is to be in a fluffy condition when installed correctly.
R-30 is usually best for 2 x 10 or 2 x 12's.
Blocking each channel with ridgid foam can affect the house's air circulation and I would seek more professional advise before I begin to do that. Let it breath!
Best of Luck,

ThePorchGuy

Fill the cavity, then use foam from a can to fill any openings.

You need the highest R per inch you can get.

Since it is not a large volume the cost should not be excessive.