Your argument doesn't really seem to touch the issue you assign yourself with (and I have no particular knowledge of Berkeley, just a scratch of something written somewhere).
I'd rather give my own argument that knowing means you have a power in foreseeing and dealing with problems. Whether this knowledge came from imagination, like mathematics which is largely made up of people's imagination, or from studying things as they occur in the environment before you, which is using ones eyes and ears etc., is largely irrelevant to knowledge. There is no singleness nor any dichotomy between knowledge and imagination because sometime knowledge is imagination and sometimes it is not.
If you'd use another word or sentence however, like "seeing is imagining", then you will perhaps enter into what you want an answer for... because, is seeing the same as imagining? Well, there may not be any simple lucid description of their differences, but by a large there is a lot of tiny differences that makes up the dichotomy. For instance, you cannot imagine pain (a reason why you cannot dream about physical pain), so pain is not imagination by principle, it is always a cause of something outside of your body. The same is tickling and other forms of direct sensation.
Now this triggers the question whether phantom-pain or other phantom-sensation is to be considered imagination or not. Because many people who loose a limb experience pain in body-parts they do not have, and by habit they may try to scratch a thumb they do not have or a foot they do not have. However, this triggers another question whether illusions are to be considered imagination or just plain deceptions; do I imagine that thing to be there or am I just being deceived and therefore not myself the cause of this thing's existence put just watching a manipulation by somebody or something else?
I could say about phantom pain that it's just a deception of your body, while imagination itself can be both a deception or not a deception. The person would normally of course find out that this isn't the case, because by studying nature, the nature of his own body, where his thumb was supposed to be. By studying this he discovers that it isn't there. In other words, it is falsifiable that the thumb exists or naught, whereas imagination itself isn't falsifiable by and of itself but needs somebody looking at it from the outside of its walls, and if somebody are looking at it like that then that somebody would by principle be knowing that the specific something isn't real. For instance, if I use my inner eye to watch my imagination's content, then obviously I exist outside of this imagination, whereas if I perceive the imagination as being in the imagination, that is, I'm not looking "at" it but "through" it, then I would not know it apart from anything else.