Is there a problem in a PK cryptosystem if the plaintexts are very small compared to the ciphertexts?
No. ​ ​ ​ ​ Also, note that by hybrid encryption, ciphertext overhead will always be

at most ​ ​ + poly(security_parameter) , ​ ​ no matter how long the message is.

For IND-CPA of any

public-key encryption scheme, or even

symmetric encryption scheme with stateless encryption

, ciphertexts for non-empty messages must be

overwhelmingly likely to be longer than plaintexts.

Your most recent comment suggests it was broken even in the following context, but otherwise you might have had a secure trapdoor permutation family whose domains are exactly the binary strings whose length equals the length chosen during key generation.