I would suggest making at least two (and maybe 3) of your mechs specialists. That could be a head-capper Marauder (a design which I think is fairly overpowered); a short-range mech that can use precision strike to kill an opponent in one turn (usually built around medium lasers); an LRM boat can work (though it's been a while since I've used one), as can a high damage SRM mech, like the Stalker that KDubya mentioned. It makes sense to have a couple of jack-of-all trades mechs (which will usually be last in line to use Precision Strike and, as such, are good with a multi-shot pilot). And then the goal is that your tactics revolve around maximizing the effectiveness of the specialists -- so, for example, if you have a head-capper Marauder that is absolutely first priority when it comes to spending morale, so you're tactics for the other mechs should minimize the use of morale ability. The short-range laser boat will probably also use precision strike but can get quick kills by jumping behind an opponent and overwhelming the rear armor.
For my own tactics I tend to have at least two mechs that are primarily short-ranged mechs, but I also tend to get tired of campaigns before I get to stage of having all assault mechs, so my experience is biased towards medium/heavy mechs.
None of that advice is specific to that mission. I don't have enough of a sense of the map to give you any advice for how to best deal with a LRM-heavy OpFor but the tactics you used of trying to take the LRM damage on a mech with 40% or 60% DR helps a lot in terms of letting you soak damage and buy time while you're killing the opponents.
I like your idea of specializing. At this point in my career I can't say that I have enough experience to know quite how to use them or which ones would best suite my style, though I'm sure I'll use some pilots as specialists as I move forward and gain more experience with different types of mechs. Thank you.