I'm the last guy to use a concentrated chem soloution. I'm not a fan of the peroxide dosing either.
For very small alge issues I don't mind this method. In fact I actually like the peroxide for cleaning coral and plugs before they go into a tank. An alge dip. So far it works quite well. I would not dip sensitive corals in a strong solution and I was advised to not dip zoas. But I have had little problem with them.
Nutrient reduction IMO will help and can work but not in all cases IMO. IME you still need manual removal.
I'm one of the reefsquad guys and I've seen a lot in the last year on this. Also in my tanks I recently did an experiment where I wound up stripping the no and Po to undetectable amounts with macro alge.
It did kill off several types of algaes but def not all and the corals actually of course did not do well.
It's a long list of macros in my 55 and actually a long list of nuscane algae too. Most folks would faint.
So now IMO the largest thing we can do is not introduce these into the system but in these cases the choice is eradication or management. But that's the choice for the individual.
fwiw. I would leave what the op has in my tank. I've become good at management and I don't mind cool algaes in the tank. I personally just trim them or use a toothbrush occasionally. On my bryopsis too.
If they get out of hand I pull them and do a peroxide dip 10 mil per 2 liter. For a few min and hit it with a tooth brush. I rinse very very well.
You can use a 5 or 10 mil on a paint brush or q tip and dab the tiny spot or even full strength and do the same. It's how I clean a coral before it goes in the tank.
No it won't effect the cycle. It probably does kill some of the bacteria on the surface but not enough to stall a cycle.
I rebuilt my 30 gal this way and dipped every rock.