That's a bit low, but 2.7GB usable is actually not the lowest I've seen when using 4GB+ on a 32-bit OS. Usually it's between 2.75-3.5GB usable. Are you sure PAE is enabled? That allows even a 32-bit OS to remap hardware address space to above the 4GB limit. Badly-written 32-bit drivers will reserve huge chunks of physical memory addresses for their memory-mapped registers.
The examples I've seen that are worse than yours usually involved multiple installed graphics adapters though, and you only have the IGP.
But yeah, I suggest installing 64-bit Windows 10 as it can use the whole 6GB, runs OK on 6GB, and is still supported with security updates until October 2025. Probably not worth investing any money in a laptop that can only be used online for another year.
After that, you could install whatever you wanted on it for use offline-only. XP runs great on only 2.7GB and your Sandy Bridge era system fully supports it.