So I have been playing through the game with no issues so far, until I got to the museum. None of the doors seem to be placed properly, and one of the strangest effects is that the hands on this clock are replaced with doors (image attached).

I have tried nuking the install and redownloading, but haven't had any success. I also tried installing 9.9 manually, but that changed nothing.

What can I do? I'd rather not start over yet again.

Both games but esp. bloodlines still have quite a number of Bugs and Glitches and without some Infos about your system, the version&mods of your game it's only guessing but there's a number of fan-made-patches that might solve your problem, the last time i needed them one could find them at patches-scrolls.
Aside from this you can try one of the mods that try to tweak the game for modern systems, most of them can be found at nexus.

And for bloodlines there's a fanwiki too
[url=http://vtmb.wikia.com/wiki/Vampire:_The_Masquerade_%E2%80%93_Bloodlines_Wiki]http://vtmb.wikia.com/wiki/Vampire:_The_Masquerade_%E2%80%93_Bloodlines_Wiki[/url]
they have a number of links but i think they didn't update it the last years.

Your bug report sounds to me like a random issue caused by too long gaming sessions. Start from a save before the museum and try again!

I kept one from just before travelling to Museum, same issue each time :(

I'm going to attempt to completely wipe the install, force it back to UP 9.6 (the version I started the save with), and see how it goes.

Re-installed completely, copied over save from just before heading to museum. No luck :( I took some more screenshots.
Ignore the top part of 3 of these, I was very lazy with creating the screenshots and accidentally overlayed them.

Oh, you should never continue an old save with a newer patch version! So rolling back to 9.6 might hopefully work...