Have you looked at clicker and target training? If they will let you hand feed them through the cage, that might be one way you can bond further with them. Instead of using your hands to try to touch them or get them out of the cage/back into it, you could aim to get them to step-up onto a wooden perch and also off it again so you can put them back inside. I have found neither of my lovebirds likes being touched or likes hands... I accept that they are like that and work around this, as otherwise they are quite happy to hang out with me outside the cage and on my body - they just really hate hands!