Expanding Key Material
Yes, you could use the HKDF-expand function to expand the bits of the key to a certain size and then use HKDF-extract to derive a key from it again. Note that the resulting key is not identical to the key you've started with. You can use the HKDF-expand again to derive one or more multiple keys with any size from the extracted key material.

HKDF internally uses a hash function (it's a Hash based Key Derivation Function). So it should be fine as long you use SHA-256 or above for 256 bit of input material. The entropy in the output material should be (close to) 256 bits.

Just like a normal hash, you would need all the input data to calculate the correct output, so if only part of the enlarged key gets known you might be still secure, as long as the amount of unknown material is large enough (so brute force attacks are unlikely to succeed).

You can leave the other arguments to HKDF-extract / expand empty if you don't know what to do with them. Using an empty salt seems fine for this kind of purpose.