Checking the integrity of RSA public key
Can I compute the only hash of the modulus for integrity?


Well, if we allow the attacker to modify the value of $e$ you use (because it's in untrusted storage and you don't verify it), how can he exploit that?  Well, the most obvious approach for him would be to modify $e$ to be the value 1; that would make generating forgeries really quite simple.

Now, if your signature verification software prohibits $e=1$ (and doesn't parse $e=-1$; likely it doesn't); well, it might be safe, as generating alternative values for $e$ where the attacker does know the $d$ value is equivalent to factoring, and there is no other known way to take $e$-th roots that's not equivalent to factoring.  This is, of course, assuming that your PKCS #1 1.5 signature verification code does all the checks; if not, then (depending on which checks are skipped) it's known to be possible to generate a forgery that passes the partial checks with $e=3$.

On the other hand, why take the risk?  It's easy enough to include $e$ in the hash, and that avoids the question entirely.  Alternatively, you can just bolt in $e=65537$ in your verification code; in that case, the attacker can't change it.