Last year I have pre-soaked with a water/peroxide solution for a very short time (minutes) to kill any fungal spores on the seeds, or in my germination baggies. I then rinsed that out and put them in the fridge for a few weeks (up to 5, I think), before pulling them out to room temperature. However, my peroxide must have been degraded as I got some fungus growing on some seeds during that incubation (including super-pricey ones) and had to throw them away as they turned to mush.
So my method was terrible.
But 5% hydrogen peroxide is enough to kill everything, including your seedlings.
The bottles they sell in the drugstore are only 3%. I am going to assume you meant 5% of that 3% solution (0.0015% final H202).
If I was you I would probably skip your hydrogen peroxide second step. Or do an experiment to compare 2 batches, one with and one without then come report back your findings.
Edited to add that my % peroxide looked wrong. That came from a Google "what is 5% of 3%". However, taking the number 3 and dividing it by 100 is 0.03 (1% of 3%). multiply that by 5 (for 5% of 3%) gets me 0.15% final that looks better to my eye. So the real calculation should be (3/100)*5. Google must be making a different calculation, most likely (0.03/100)*5. The joys of trusting Google.