xosm files have absolute paths?
Absolutely not. XOSM files have relative paths. PixInsight projects are completely self-contained and modular by design.
On my desktop it opened the target file fine but not on the laptop.
I can't reproduce that. To check if everything works well, I have just opened on this computer (Linux workstation) a complex project (150+ images and lots of process icons and text files) that was generated on a Mac OS X machine, which has a completely different filesystem. I've opened a copy of the same project on a Windows 7 x64 virtual machine. In both cases everything opens fine and works without problems.
We perform the same action you're describing all the time and have absolutely no problems. I regularly move big projects from home to several computers on our university, and Vicent uses huge projects to give talks and in critical tasks of his job everyday.
Have you copied the *entire* .data folder along with the .xosm file? Could it be that there are some missing files in the .dat folder? In such case your project would load only partially.
Something about HDRWT not being available.
Hmm, that shouldn't happen (and doesn't happen; I've just checked) with the latest version of the MultiscaleProcessing module. For compatibility with old versions of this module, the HDRMultiscaleTransform process has "HDRWaveletTransform" as an alias identifier, so the transition should be transparent. Please check that you have the MultiscaleProcessing module version 2.3.3.186.