The Master System's VDP is a modified TMS9918, and so most Master System games run in its extended 'Mode 4' setting. That was the only video mode, therefore, that I'd emulated in any form.
For some reason, the older computer games are, the more charm they seem to have to me (maybe because the first games I played would have been on a BBC Micro, which certainly looked a lot more primitive than the Master System games I've been attempting to emulate thus far). I dug out TI's TMS9918 documentation - the differences are quite significant! Tiles are monochrome (though you can pick which foreground and background colour is in use - to some extent), the palette is fixed to 16 pre-defined colours (one of which being 'transparent') and sprite sizes and collisions are handled differently. Various features found in the Master System's VDP (scrolling, line-based interrupts) also appear to be missing from the 'vanilla' TMS9918, but I'm not sure whether or not they make an appearance in the SMS variation of the VDP or not, along with the original TMS9918 limitations.
At any rate, the emulator now has a 'SG-1000' mode. The only differences at the moment are that the TMS9918 palette is used and line interrupts are disabled, so you can still (for example) use mode 4 on it.
From first to last: Drol, Choplifter, Hustle Chummy, Championship LodeRunner, Space Invaders, H.E.R.O., Elevator Action, and Zaxxon.
All but one of the SG-1000 games I had ran - and that was The Castle. According to meka.nam, this has an extra 8KB of onboard RAM. Whilst doing some research into the SG-1000 and the TMS9918, I found a forum post by Maxim stating "The Castle runs nicely on my RAM cart :)". Enter the new ROM mapper option to complement Standard, Codemasters and Korean - it's the RAM mapper, which simply represents the entire Z80 address range with a 64KB RAM.
That seems to have done the trick!
I mentioned that the palette was different in the SMS VDP and the original TMS9918 - here's an example (SMS on the left, SG-1000 on the right):
I'm assuming this is the result of truncating the TMS9918 palette to the SMS 6-bit palette without needing to implement two palette modes on the new VDP. Another comparison is between two versions of Wonder Boy - SMS on the left again, SG-1000 on the right: