I tried searching for 0156 in Agner Fog's table. Some instructions aren't exactly what you asked for, but seem worth mentioning.
I know you wanted to exclude mov type instructions, but movsx r32, r16/r8 is definitely not eliminated, and definitely runs on any of the p0156 integer ALU ports. Similarly movsxd r64, r32. Only mov r32,r32, mov r64, r64, and movzx r32, r8 can be eliminated (0 latency, no unfused-domain uop).
If you were ruling out movzx/sx because of possible mov-elimination, look again at movsx. It may be the only such instruction.
bextr r,r,r is 2p0156. But it's probably actually p06 + p15 or something, implementing it with something like shift (p06) + BZHI (p15) uops. That hypothesis can be tested by mixing it with some shifts or p15 instructions.
xchg r64, r64 is 3 uops for p0156. According to my reverse-engineering, I think each uop is a reg-reg mov that's not subject to mov-elimination, and actually needs an ALU port. One of the registers involved is an internal microcode-use-only register that's not architecturally visible, but does participate in register renaming. (I think we have other evidence that there are a few extra logical registers that don't have an x86 name, e.g. using up PRF entries). But of course neither destination of the whole x86 instruction is write-only. leave also has 2p0156 (possibly not using the stack engine).
salc is 3p0156 (set AL from carry: undocumented, not 64-bit mode) but that's probably sbb same,same and a merging uop into RAX. So it's probably like lea r16, [m] or imul r16, r/m16, imm or movsx r16, m8 that also have a merging uop into an architecturally write-only destination.
movbe r64, m64 runs on 2p0156 p23 on SKL. But movbe r32, m32 runs on p15 p23 so there's probably just one extra p0156 uop in there, or a p06 uop. bswap r64 is p15 p06 so we can be pretty sure that's what movbe uses. I assume movbe r64, m64 is really p15 p06 p23, i.e. load + bswap, but Agner didn't manage to pick that apart.
So other than movsx and movzx dst, r16, mostly this answer is debunking / ruling out possible p0156 instructions from Agner Fog's table.