While agree that LMS is the most likely name for the branding, I don't think it is appropriate at all.
London Midland was only 2 years ago, and is extremely 2/3 the same as London Midland Scotland. Given what a mouthful to say "London Northwestern Railway" (8 syllables) and "West Midlands Railway" (6 syllables) me and a few others I know still accidentally call the services "London Midland" still, (a vastly shorter 4 syllables). Given that LM was the local stoppers, and LMS will be the express trains, I think that is even more so confusing, especially to those who don't take the train very frequently. (One could assume LM just extended to Scotland if you weren't a West Midlander).
Given WMR/LNR still use the LM code, even more confusing!
I think something of 2000s/2010 nature which the WCML skipped from using brand king virgin instead would be more appropriate.
e.g.
West Coast Trains (<region> trains)
West (Coast) Express (Wessex Express)
Central Trains (It is certainly the economic centre with the large cities of London, Birmingham, Manchester & Glasgow comparative to the other North to South franchise), or perhaps go))) more modern "Central Railway"
Just "West Coast" (Too obvious? East Coast)
Great/Grand West Coast Railway (GNER/GC)
"West Coast" or "Central Railway" would probably be my favourite.