Definitely remove the straw; you want to give the seed something to grow in, not on.
The straw would still be there when the seed in the compost is trying to grow and the straw would be too loose ontop of the soil for it to root properly.

The compost over the soil will help retain moisture but maybe you could think to add a lot more organic with the compost; it would be dug in or rototilled in, then the grass seed.
If the present soil is, as you say, it does suggest it is very loose, has no organic matter in it to hold moisture and feed the seed.
For that then I would suggest the added organic matter be dug in to a depth of 3" - 4" before seeding.
The straw itself would have to be aged well because it too could drop seed and you'd have a meadow instead of a lawn.