How do I recover an "invisible" OS X boot option in BootCamp (running on a 2013 MBA)?
I managed to run more diagnostics against the disks from the Windows side. From the BitLocker password screen, choose something like "Esc", "Esc", "Skip this drive", "Advanced", "Utilities", "Command-line" (please correct), then use manage-bde -unlock to unlock the BitLocker drive, and use various wmic commands to examine the disk partitions. Through this I confirmed:


disk0s2 is my OS X partition. There are some tools which can change Guids in the GPT; it seems this is what happened to my OS X partition. This explains why OS X doesn't recognize it, but when booted into Windows I could access the data in it no problem.
disk0s4 is my Windows partition. To the best of my knowledge, the Invalid signature in fsinfo block errors I was getting are due to it being fully BitLockered which means the encrypted bits on disk don't look like a valid partition. Windows can apparently recognize the signature and warn you on trying to read from a locked BitLocker partition.


These are all the possible solutions I came across (largely through Apple's very helpful Tech Support), including the things that didn't work for me and why:


Backup your existing partitions as images to an external drive via the "New Image" option in Disk Utility. I do have a 1TB backup drive, but it's NTFS formatted and apparently OS X can't write to NTFS disks by default (try this to enable NTFS write access on a standard OS X installation - but it doesn't work in Internet Recovery as the terminal gives you access to a read-only file-system only).
Backup your existing partitions as images to an external drive via [name of Windows program here]. I couldn't find any built-in Windows utilities capable of this, and due to my Windows login being disabled I couldn't access a full Windows environment to run any proper backup tools. I tried running them from the BitLocker command-line but I couldn't get any of them to work, I assume because the environment is heavily cut-down (and most of them insist on "installing" rather than just "running").
Backup your existing files by transferring them to another Mac via Target Disk Mode - I have no second Mac on hand to make this work.
Just run Repair Disk - it is reportedly very unlikely to damage anything. When I tried this it just said it couldn't repair the disk - possibly because the BitLocker partition doesn't look like a valid partition (see above). A bit of an anticlimax after agonizing over the decision for days...
If you have the Invalid BS_jmpBoot in boot block 000000 error, someone wrote a utility which can reportedly fix the GPT up again. I couldn't use it because I couldn't find a way to install all it's dependencies (e.g. Python...) in the Internet Recovery read-only file-system.
Just erase some/all partitions and start again - in the end I decided to take the plunge and erase the existing corrupted OS X partition, leaving the rest untouched (once I'd made myself relatively certain which one was OS X). Disk Utility makes it very straightforward to erase a single partition and install OS X on it.


Erasing the OS X partition and re-installing has now resulted in me being able to dual-boot OS X and Windows 8 via BaseCamp as desired. Next time I'm in Windows I'll have to check whether I do indeed have "MacDrive" installed, and remove it if I do.