Looking at the way that plate has broken, something's pushed it out from the inside. It covers the centre diff lock mechanism. Yes you can certainly loosen the front nose cone of the transfer box by taking its bolts out but can't withdraw it completely because the gears in the main bit of the transfer box are in the way. So it's usually easier to separate the transfer box as a whole from the primary gearbox and work on it comfortably on the workbench.
It's difficult to imagine why it put the hazards on, as that's all mechanical rather than electrical. However, the ECU is likely to be monitoring crankshaft speed, road speed (via the rear transfer box output), whether it's in centre difflock and whether it's in high or low range. So it maybe that these values got out of range momentarily as the problem happened. Similarly, I've suffered from both low oil and bearing wear at various times and they don't break the casting all by themselves. Some foreign object has come into contact with the mechanism and got thrown out through the side.
Anyway, try the garage first. In a career of Land Rover ownership, it's likely that you will have the transfer box out one day. But this is so soon after purchase that you want to try the garage - there's lots of policy and legislation in your favour.