My small pond is 20 years old and has flagstone around it. Over the years I have encouraged a native moss to grow around the stones. When I pull broad leaf weeds , they pull up the moss. Can I poison the weeds and not hurt the moss? Some of the weeds are ; houttuynia, violets, money-wort, all very invasive , and a few others like taro and aspidistra that are not much problem to pull.
Most systemic herbicides (including glyphosate) do not kill moss. Moss is a "non-vascular" plant, so herbicides which work by getting into the plant's sap and killing its roots have no effect on moss.
Just take care that you don't buy a mixture of glyphosate with something else which does kill moss, algae, etc. "Roundup" is advertised as moss-safe.
If you want to be cautious, try it on a small area first, but it should be fine.
Note, while systemic herbicides do not kill moss, many products sold as "moss killer" will kill everything, not just moss!
I know what you are going through. Pulling baby weeds from the moss will tear up bits of it. Think of it as renovation and aerating? I got a little OCD pulling baby weeds out of gorgeous moss beds. Even beds of woolly thyme.
You could use chemical gloves and a small sponge soaked with glyphosate that sits in a little dish with a dry cloth beneath so absolutely not a drip will drip...wipe a few leaves of the weeds and they will die, roots and all. Takes about 3 weeks.
Way easier to just grab a cup of tea or a glass of great wine and sit and pull baby weeds. I say baby weeds because big weeds will cause a lot more carnage of your pretty moss bed. Big weeds will have already seeded to boot causing more weeds soon.
The bigger problem or rather more difficult task is maintaining the perfect environment for the moss. Grass dislikes the acid soil and shade. Grass as one of the weeds most common in a moss bed. Plucking baby weeds out of your moss bed is worth its weight in gold for stress relief. Just sayin'. I loved going out on breaks to pluck erroneous plants out of the gorgeous woolly thyme bed. There is no better treatment. Win - Win?
Houtynnia (arghhh sp.) is major invasive. Best to use a thin barrier of plastic or galvanized metal around that mass of plants. I would use glyphosate as I've described for the houtynnia outside of that barrier. Or just sit in your special space and pluck. Sounds like a wonderful spot, blacksmith!