You wouldn't add chlorine gas for the reasons you've given- you'd add bleach solution instead. Bleach isn't benign, but people handle things much more hazardous than bleach every day without incident. Bleach costs more than chlorine, so if you need a massive amount, you'd be stuck with chlorine.
You make a mole of hydrogen for every mole of chlorine when you electrolyze water, and if you do the math, you'll know that you would need to add roughly as much of whatever hydrogen-destroying additive someone could imagine for you, as you would add in bleach if you went the non-electrolysis route. So even if the idea wasn't dead from a chemical reactivity basis, you wouldn't be adding "a tiny amount".
Any separator you add to remove H2 would need to be downstream of the electrolysis point a bit, to ensure that you don't also vent chlorine. It does take at least some mixing and RT for the chlorine to dissolve, even though in a mixed cell you're making OH- ions right next door to dissolve some of it as hypochlorite (Cl0-).
You're not making enough hydrogen to affect the heat transfer between the plates of the exchanger as it flows by, unless you're circulating in a closed system without any vents at all... Unless you're massively over-chlorinating, the volume fraction of gas generated in a single pass in order to do adequate breakthrough chlorination is tiny. The hydrogen only accumulates at high spots because it can do so over long periods of time. If your exchanger is designed such that it has a dead zone at the top which can fill with gas, it's been designed wrong. And if it is being operated at velocities low enough to be unable to sweep out gas bubbles, it's been designed wrong.