Bigger is pretty much always better. The only real way to force smaller mechs into later stages of a game would be, as others have already noted, to have tonnage limits on missions. Megamek, if I recall correctly, restricts generated missions to certain lance weights. So if you're running a Megamek campaign, you can't just deploy all assault mechs all the time. You have to have a mixture of lances of different weights to fulfill various contracts.
Don't know if this game has it. It'd be nice if it did. Otherwise, you'll have to have self imposed rules. One of the table top books, Campaign Operations, describes bonuses for certain lance compositions and gives those lance types special names (Force Lance, Command Lance, Fire Lance, etc). I was thinking I might try to always honor those restrictions so my lances are somewhat more "thematic" rather than just mix/max optimized.