"can't do anything except eating" vs. "can't do anything except eat"
It seems the answer (in part) is found in the modal verb can; modal verbs (can, could, may, might, must, shall, should, will, and would) accept only bare infinitive verbs.
A. My dog is so lazy. It can't fetch sticks, chase after a cat or even roll over but it can eat.
- My dog is so lazy. It can't do anything except eat
If the OP substituted can with any of the preceding modal verbs, the verb that followed "except" would still be the bare infinitive.
- My dog is so lazy, it won't do anything except eat.
However, if we switch the modal verb (can) with a normal verb that accepts a gerund, we have the following
My dog is so lazy, it enjoys nothing but eating.
My dog is so lazy, it enjoys doing nothing except eating.
The version with the bare infinitive works less well, it is less harmonious than the former but I wouldn't say it was "wrong" per se.
My dog is so lazy, it enjoys doing nothing except eat.
Resources
There appears to be precious little on this aspect in the grammar books which I consulted at home. The advanced English grammar books tend to dismiss it entirely whereas the student-friendly books tend to repeat the same rule.
When except is followed by a verb, we usually use the infinitive without to.
I couldn't do anything except just sit there and hope.
She did nothing except complain the whole time she was here.
Except is normally followed by object-forms of personal pronouns (me, him, etc.), except in a very formal style.
Everybody understood except me.Practical English Usage by Michael Swan
and in A Practical English Grammar by A.J. Thomson and A.V. Martinet
98 B The only exceptions to the gerund rule are except and but (preposition), which take the the bare infinitive:
I could do nothing except agree
He did nothing but complainHowever, if but is used as a conjunction, it can be followed directly by the full infinitive or gerund:
Being idle sometimes is agreeable, but being idle all the time might become monotonous.
To be idle sometimes is agreeable, but to be be idle all the time etc.