CFOUR + MRCC
You asked about MOLPRO and Psi4. As recently as 2021, DBOC (diagonal Born-Oppenheimer correction) wasn't implemented in Psi4. The same is true for MOLPRO. No other types of Born-Oppenheimer corrections are available in those programs as far as I know. You mentioned that CFOUR can only do DBOC with CCSD, but with the MRCC interface you can actually do CCSDT, CCSDTQ, CCSDTQP, etc. (though these will not be helpful for the HD molecule since it only has two electrons!).
Here you can see my input and output files for a DBOC calculation with CCSDT using CFOUR with the MRCC interface.
For the carbon atom with the aug-cc-pCVTZ basis set from the EMSL Basis Set Exchange (circa 2018), this was the result of my DBOC calculation with CCSDT:
The total diagonal Born-Oppenheimer correction (DBOC) is: 0.0017104162 a.u.
The total diagonal Born-Oppenheimer correction (DBOC) is: 375.392974 cm-1
The total diagonal Born-Oppenheimer correction (DBOC) is: 4.491 kJ/mole
If you explore my AI Energies database (a database of completed ab initio calculations including the relevant input files and output files), you'll see that I frequently did DBOC calculations with CCSDTQ and even beyond that.