Easy enough to pull up the grass, so no need to remove the soil. I would think that it would not have developed a deep root system yet.

If you don't want the grass you don't need to tip it all out. The soil is still quite soft so you should be able to just pull it out a tuft at a time. There isn't very much. Just shake off the loose compost back into the bed and throw the grass away. Just wondering why you planted bird seed in the first place? Keep bird seed for the birds and flower seeds for the garden.
Bird seed is a real mixture of seed that will grow into many different kinds of plant that would not flourish if dumped onto the soil so close together anyway.It may not even be suitable seed for growing outdoors in Britain. eg peanuts don't grow outside here.
If you want advice before you do your next job why not put a question or two on here and there will always be someone around to advise you how best to proceed.
And next time you sow seed sow it thinly. If you just dump it on top of the soil it will never come to anything.

bird seed has wheat/corn rather than grass so you could leave it and save the seed for feeding the birds later. they are shallow rooted so pulling up is easy.

welcome to GoY too.

I got some Phacelia once from birdseed. I reckon that was a bonus!