What info should be signed with a digital signature to provide entity authenticaton for DH?
Typical signatures start by hashing the message. Thus including more data in a signature doesn't increase the size of the signature, if the recipient knows what you signed (in this case, remembered what they sent in step 1).

You need to hash more data if you include $g^x$, but the cost of hashing a public key is quite low.
If you didn't include some kind of challenge value in the signature a compromised session key would allow impersonation in future sessions, which is undesirable.
I recommend signing all the data exchanged so far (transcript hashing). This leaves much less room for an attacker to manipulate an exchange, which is important, especially for complex protocols like TLS.