I'm trying to optimize my signature and verification scheme for an embedded device and I'm finding race conditions at just 0.5s/verification. Instead of making the device compute the SHA-256 hash of the data, could I just use an AES encryption and sign that with PSS to accelerate the process, or does it need to be a hashing algorithm?
In general, RSA-PSS requires a hash algorithm for the mask generation function and other operations, so it's doubtful that you could actually make RSA-PSS work in any functional way with something that is not a hash algorithm.
The idea you're proposing is also likely insecure, so even if you could get it to work, it wouldn't be effective as a signature scheme, since it could probably be forged. That's because AES, unlike a hash function, allows users to invert the operation (that is, decrypt), so an attacker who knows the key (which, if you hard-code it, they do) can likely create arbitrary messages to sign.
For a secure digital signature, you really need a secure hash algorithm, which means that you need something like SHA-2, SHA-3, or similar (MD5 and SHA-1 are not secure and should not be used). If possible, I would investigate a possibly more performant SHA-256 implementation here. You could also try the hash algorithm BLAKE2s, which is both cryptographically secure and much faster in software than SHA-256, and may meet your needs better.