A boring 4-high solution. I'm 100% confident that P3 is possible, but I don't know enough to get anywhere. You can't let go at P3 except for a debonder or if you somehow manage to get at least one atom of whatever you're holding to be completely stationary for the cycle before your drop operation (including rotation!), but something 3i-like should be workable. Since we only need a P3 instruction tape, you can use multiple "3i" arms to hopefully simplify things by a lot. At P4, you CAN let go, but I still don't know enough to get anywhere at height 4. Let's see what the masters come up with.
I've edited these notes several times already, but this one is important enough to mention as an edit: Period 2 may just barely be possible if you wand EVERYTHING. Having a 3-atom input is really nice, since you can grab then move. This uses up your two instructions, but you can eventually move over a debonder to "drop" one or two atoms. One atom will necessarily become "waste" that will never be outputted, but it could still be used as a wand in theory. The important thing here is that a period 2 INSTRUCTION TAPE doesn't mean period 2 MOVEMENT. To make mu point, I would like to refer you to the MANY 3i solves that have shown up for various puzzles. You have very limited track space since arm length can only go up to three, but clever wanding should allow you to "schedule" individual manipulations of broken-off atoms. I cannot even begin to imagine what this would look like, but I am 100% sure that P2 is possible. I am less confident that it's possible in h4, but I'm waiting for the top players to prove me wrong on my doubts.