The bits field can only change every 2016 blocks (4 bytes per header),
the timestamp can not be less than the median of the last 11 and is
usually only a small amount over the last one (saves 2 bytes per
header), the block version is usually one of the last few (save 3
bytes per header).

But all these things improvements are just a constant factor. I think
you want the compact SPV proofs described in the appendix of the
sidechains whitepaper which creates log scaling proofs.