They don't look to be insanely close to the edge. There are a lot of factors that could affect affect blow/bulge out. Centerline to edge distance, material thickness/hardness, hole size/form, hole method creation (localised hardening), anvil shape, press pressure etc. A harder grade of self clinch studs are required for stainless as the studs need to be harder than the installation material. That shouldn't be a problem for your fabricators though.
I see you're using 1.5mm sheet which is what I would go for.
Stainless is harder than mild steel which has a double edged effect. It's harder so it's less likely to bulge but it's also more likely to bulge as more force is required to install the self clinch studs.
There's a load of good information on self clinch inserts [here] if you want some bedtime reading.
Centre-line to edge challenges are highlighted on page HB-12 of the manual and there is some information on page HB-11 too. There is a link to a tech sheet but sadly the link is broken.
I would speak to your fab guys ( ) and mention the bulging. If you are going to go with stainless you could ask them to try doing some bulge tests ( ) on some scrap material to test different pressures etc before committing to the production run.
Edit: I meant to say: I would stick with the self clinch studs if you can get them installed correctly, they give a much nicer finish than welded studs IMO.