I was recently given a bespoke system built around an MSI MS-9803 (V1.X) Mainboard (mini ITX). The processor for this system is an Intel® Core™2 Duo Mobile Processor T5750.
I purchased a new case but the bespoke passive heatsink does not fit. Any ideas?
I don't mean to sound facetious, but your best idea is to modify the case so the heatsink fits. Socket P is a mobile socket and thus has no standard heatsink mounting pattern because laptop makers all use custom (and frequently different) cooling solutions with different hole spacings to fit their custom-made chassis. That's why your motherboard came with a heatsink, which you correctly noted was a custom, made-to-fit unit.
One thing you could do if you are pretty comfortable with metal working is to fabricate your own backplate and mounting plate to fit the MSI board's mounting holes but have the mounting plate also be drilled and tapped to accept a standard bolt-through heatsink mounting pattern (e.g. LGA771) or a standard heatsink mounting bracket (e.g. AMD Socket AM2/AM3). That would give you access to a very wide variety of heatsinks of all shapes and sizes, but it may not fit and it takes some skill to fabricate such a plate.