To invite or not to invite?

The general rule is to invite in 'groups', so, for example, if you invite one cousin, then you invite them all. However, I think you can get away with it when there is such a huge gap between the contact you have with one sibling and the contact you have with the other two.

It's a bit more problematic if you invite sibling 2 but not sibling 3 as there is not such a big gap in the kind of contact you have with them. There isn't a huge amount of difference between birthday/Christmas cards + an occasional text and Christmas cards only. So in your place, I would probably invite either just sibling 1 or all three. But it really is up to you - you can invite whoever you want to, you just need to be prepared for possible fallout!

If you are unsure and you are close to sibling 1, why not ask her what she thinks?

I don't have siblings, but I do have uncles and aunts. I invited 2 aunts and one uncle whom I see multiple times a year - with the others, my only contact is the annual Christmas card and I hadn't seen any of them for a good 5 years before the wedding. I still had complaints from one aunt that she was deeply hurt because I hadn't invited her, but since she'd turned down invitations to meet my OH before we got married, I didn't feel too guilty. (We didn't want anyone at the wedding whom we didn't both know)