If your peppers are fine until the transplant, it could be that they're just not mature enough. Three inches tall doesn't sound very mature, if you want them to get very big or be productive. It could also be a deficiency in your soil. I recommend a soil test. They might need more magnesium, manganese, nitrogen or something. They might need less nitrogen. Knowing exactly how they die would help, like if the shoots shrivel, or if the leaves turn yellow, or if they wilt. It could be an insect pest.
It could also be a pepper disease that your tomatoes are resistant toward.
It could also be that the sun is shining too brightly when you transplant them, especially if you plant them later in the season than a lot of people do. That could make them wither and die if you don't shade them. You might try planting after the sun starts to go down, or shade them for a couple days.
If your peppers are looking sad before you transplant them, I'm guessing your peppers just aren't getting much light. You could put them under a table lined with mylar blankets and put some CFLs in there, both 6500k and lower (lower ones help against damping off better). Or just give them an overhead light. That might be slightly easier and cheaper.
I also recommend giving them some potassium sulfate approved for organic gardening, unless the stems are already super strong, which could indicate they have enough. If they're soft and bend easily, or ironically, if they're getting bark, they could use more potassium. They'll get strong fast if you give them that. It should only take a few days. Tomato stems get strong faster with potassium sulfate. Potassium, if you don't have enough, after application, will strengthen the plants, help protect against disease and insects, help the plants grow faster, and increase fruit size, among other things. If you give them too much, they may get nitrogen deficiency, but it's a lot easier to give a pepper plant too much nitrogen than it is too much potassium, unless you're applying ashes or something, in which case, you don't want to add heaps of them. I recommend avoiding ashes unless you're experimenting, and know how much and what kind to use. Don't listen to people who act like potassium sulfate is a lot more expensive than harmful kinds of potassium like potassium chloride. It's not that expensive, although it is really somewhat more expensive than potassium chloride.
I don't recommend just using an NPK fertilizer with potassium in it. In my experience, they don't seem to add as much potassium as is helpful, or the other nutrients overpower it somehow. Just pure potassium sulfate should work fine, unless you've got a lot of potassium in your soil and don't need it anyway. I've been using a tablespoon or a half tablespoon per gallon of water, but a teaspoon is the recommended for deficient potted plants, according to someone who answered an Amazon question about it. This is the kind I got. My plants seem to enjoy it. This will initially make your plants drink more water for a couple days, though, but it is supposed to help with drought tolerance.
Along with Wayfaring Stranger's answer, I have also heard that peppers love Epsom salt. I haven't verified this yet, but I have given them a little before with no negative consequences.