Taking 1 or 3 levels of Soldier and then going drone mechanic can work out well. With a one level dip, you may have a little pain at level 3 or may have to take weapon spec as a feat at 3 and then retrain at 4, but otherwise, you'll be fine.
Taking 3 levels of drone mechanic with a combat drone and then using that drone to be an explode heavy weapons platform and going soldier the rest of the way would also work pretty well, I would bet.
I don't think Exo Mech and soldier mix very well as they double-up on some stuff, Exo-Mechs are already pretty soldiery. But I've been wrong before.
The scenarios are (mostly) designed for any random group of players to be able to play in them, so you don't have to be especially optimized either as individuals or as a group for most of them. Depending on how you go there may be a couple where you might feel left out for part of them, or you might be behind if you built especially poorly. But I'd argue that they're the exception and not the rule.