Why does RSA signature need padding? Actually, I think I know why, but consider RSA2048 and SHA256. What about to, instead of padding, just use pbkdf2 with just 1 iteration to expand the 256bits hash to full 2048bits and use those 2048bits as an input for the plaintext RSA?

What would be wrong with that crypto-schema?
As indicated in the comments, using PBKDF2 instead of e.g. PSS is replacing the padding scheme with another padding scheme. PBKDF2 is used as a key derivation scheme that accepts a password. Although that password needs to be encoded as bytes first, it does mean that you use PBKDF2 differently from how it is intended.

If PBKDF2 is used without a salt then you would have a deterministic scheme. If you do use a salt, then you'd need to somehow put it in your signature scheme. That's not necessarily a huge problem, but your current scheme is not fully described at best.

Similarly, you probably simply mask out the most significant (leftmost) bit of the PBKDF2 result to zero. That's fine as long as you describe it correctly and perform verification correctly. It's slightly awkward that the result is within the range $[0, 2^{(l - 1)})$ rather than $[0, {N - 1})$ to create a full domain hash but I guess we can look past that.



PSS is provable secure (given the usual pre-conditions, such as RSA and the used hashes being secure of course). Your padding scheme on the other hand lacks any kind of security proof (although, as SEJPM already commented, it would probably be easy to show that it generates a full domain hash, and that's considered secure). I don't see why it would not be secure - I think it is even likely that it is rather secure. However, with any cryptanalysis on a fully described scheme, that's just an educated guess.

The question is why you would need it since PKCS#1 v1.5 padding (for signature generation) only requires a prefix / postfix in addition to the hash value and PSS can be configured to use a single hash function (or any combination of two secure hash functions, one for the data and one as configuration parameter for MGF1).

If you'd study PSS you'll probably conclude that it is not that different from your scheme at all...