If you say 210, are you going to absolutely refuse any offer that's less than that? If you would, then you picked a bad number: you should have said 220. Or 230. Or 250. You've already hypothesized that the manager's actual range is 200-250; given that hypothesis, you should be targeting at least the top of that range, not the bottom 20%, if you're going to give a number at all. And if your objection to that is that you don't know the manager's range, yes, that's the point! That's why you shouldn't be risking anchoring the conversation in the wrong place.
If you wouldn't refuse an offer less than 210, then the manager is not going to lose the placement by negotiating with you for less. Yes, 210 is "within budget", but that doesn't mean negotiating for less has no benefit. That's still budget he could now use for something else.
In my experience any number you give is not going to be interpreted as your absolute minimum. It's going to be interpreted as the first step in a negotiation--one in which you should have waited for the company to take the first step, but you didn't and the company will feel no compunction in taking advantage of that.
But in any case, even if you get 210, you have, as I said, foreclosed any chance of getting more. See below.
> You also might get less.
But you're not going to get more, and that's what you should be thinking about.
You've already said the manager's actual range is 200-250. If you say 210, then, as noted above, you're restricting yourself to just the bottom 20% of that range no matter what.
If you give no number and the manager offers 200, when his range is 200-250, that probably means he views you as a baseline hire and you wouldn't have gotten 210 anyway if you'd given that number first. In any case you're no worse off not giving a number; the key point is that the manager views you as a baseline hire.
If you give no number and you're a good enough hire that the manager would give you at least 210, he'll probably offer 210. And then you can negotiate from there.