When you are writing a paper, your task is not to prove something. It is to communicate to others that you have proven it. If you make some theoretical mistake in your paper, that's your problem and you should correct it. If you write a paper which does not communicate your results well enough and misleads your readers to false conclusions, this is also your mistake, and it is your job to correct it.
And yes, writing is hard. We love to jot down our train of thought in the way which makes most sense to us, and then we expect that others will understand. In reality, a good paper tailors its writing to the readers. The best authors design their text as a track for the reader's train of thoughts, knowing the possible junctions of misunderstanding and gently leading the reader away from them.
The average author, and sometimes even the best author, cannot write a perfect text from scratch. Readers will misunderstand things every now and then. And a reviewer is a very thorough reader. If your reviewer misunderstands something in your paper, then it is likely that a large swath of later readers (who don't try to follow your text anywhere as closely) will misunderstand it in the same way. This is a major flaw in your paper, and has to be fixed. Luckily, such fixes are very easy with a bit of thought.
In your case, the problem is obviously that the assumption you made was not salient at the time the reader reached your proof. You have to change that. For example, in the place where you list your assumptions, add a sentence or two discussing why the assumption was made and what are its consequences. It will force the reader to think about the assumption and notice it, and keep it in mind for the rest of the paper. Should he think of his counterexample later, he will notice himself that it is not a counterexample, as long as it is obvious how it hurts the assumption. If it is not obvious, it is a good idea to discuss (somewhere after the proof) how examples of this type are not counterexamples, because there is a twist which makes them hurt your assumption.
After the paper has been corrected in this way, you can decide whether to resubmit to the same journal (point out to the editor that it was a misunderstanding and you have made changes to the text to clarify the point) or to another venue.