I have used tips, beams, and everything in between for making overlays. When you saw slabs of the beam, the underside will be spongy material. That's not altogether bad, because it can allow for the glue to penetrate and get a mechanical grip on the horn. I tend to use 5 minute epoxy with overlays. My overlays tend to be fairly thin, so there is not a great deal of the spongy inner part left, but even still, I make sure the horn is warm to the touch and laying inner side up on the bench and sometimes I thin the epoxy very slightly with acetone to make it more runny so that it penetrates deeper into the pores.
Gel type cyanoacrylic "superglue" works well, too.
On one of my earlier overlays with horn, the string grooves cut into the spongy material and I dabbed it with super thin cyanoacrylic. It kept soaking it up a little at a time until it was filled. At that point, I left it to cure. No problems.
