Under a tree is a perfect spot to put bulbs because the bulbs grow in the early spring, before the trees leaf out.

Yes, birch trees are "thirsty" but they need water because they lose it through evaporation from their leaves; in March/April they are not covered in leaves yet, so there will be plenty of moisture int he soil.

In the mean time you have to have to supply a "thirsty" tree water and protect the soil with mulch - you may as well plant flowers in it to take advantage of what's there!

As noted planting under the tree is not a problem, but planting in a lawn that will need mowing while the daffodils have all their foliage is problematical. If it is a "no mow" lawn it will work. Al

If you are happy with long grass until the foliage dies down there's no problem. You see lawns with long patches in early summer all the time here where planting bulbs in grass is standard practice. It will look fine as long as it is obvious it is deliberate rather than just an oversight.
And why stop at daffodils? You could start the show with winter aconites, move on to snowdrops, then crocus, then daffodils and finally species tulips.

Here is a link that might be useful: Bulbs under trees

Thanks, everyone! The lawn is indeed low-mow, so it sounds like this plan will work after all. I'll race out and get to it today!