My kids are 9, 8, and 8. I am, for many reasons, hypersensitive to my kids picking at each other, the bickering over stupid stuff really shoots my anxiety sky high.
However, as you say, you can help give your kids the tools to deal with each other, but you can't MAKE them use them. And I think sometimes that what bothers us really really really doesn't bother the kids (which I think you get too, since you're trying to tune them out a bit!).
I don't know if this will work for you, but I've found it helpful during the cranky hour(s) of the afternoon or when the kids are doing their own thing to put on my headphones and listen to some amusing/fun/informative podcasts (it helps, for me, to focus my attention better than music). The headphones muffle the sound of the whining, but if I was really needed, the kids could certainly get my attention. I become more relaxed, because my mind is engaged in something enjoyable, rather than fruitless worrying and anxiety about my worst imaginings for the kids' future (that they'll hate each others' guts--which I know is not true, they might fight and bicker all the time, but if anyone else messes with their sibling, they go mama-bear, still like to pile in the same bed to read, ect.).
There have been a few times where the volume increased to the point that I took off my headphones and went to intervene and the kids actually SAID to me, "Mama, we're okay, go put your ipod back on." and they actually are getting better at managing the bickering volume.
I would not suggest this for young children! But at 8+, I think it's okay to give the kids some bickering/pestering space. If they're actively involving you most of the time, that's another issue, but it sounded like tattling is not happening ALL the time and it's mostly your anxiety and need for them to "get along" that's causing you stress. I can't ignore and give space without something ELSE to listen to, to be honest. Have you tried that? I used to try to listen to the radio playing in the air, but I find I really need the muffling of the earphones and the extra sound lets me be far more tolerant.
