Balancing probs with fixed teams

If a game is balanced, then there are always "good" units.
However, I think you mean that there is no RPS system implemented. Meaning that one player cannot make a bad choice from the start.
Or, each unit has a unique role in the army.

I assume that players play open new options? But are limited once again playing in a new round, against another player?

You could post your idea's here. There are plenty of people here that can help.

...our game is quite similar to what you have described. "Armies" are fixed, it's chess like in flavor and you don't have to use a resource system if you want to jump straight in.

Getting the balance right without resources I found much easier than with, getting the balance right among the troop types was harder, and getting the armies as a whole balanced with each other whilst being very different (and fun) being the toughest part.

I found the trick is not to have any one unit that acts as a keystone for the others and that each unit can be effective on it's own.

Adding resources does allow for some more interesting abilities and can make it much more tactical, but also make the game much heavier and not as 'new payer' friendly.

I'm intrigued that you have a system that allows you to buy troops, yet each player's armies are still 'equal', that's where you could find your biggest balancing problems as you'll have to make sure that there are no combos that give any one player the edge.

I've been playing two Games Workshop classic games' campaign mode for some time, and that campaign aspect is the one that needs a lot of attention to be balanced.

Blood Bowl balances the amount of units in a team so that a player may only wield X amount of them, but may own more. And the really unfortunate players can have free units for the duration of the match to fill the quota.

Mordheim on the other hand doesn't give a shit about underdog players, and it kinda shows. Many a time players do want to start their whole team from the start rather than trying to cope with the losses they suffered. The game tries to balance things by giving the underdogs more money (less upkeep). There's also a system that lets you bail out before things get too bad, but usually due player stubbornness matches are fought to the bitter end.

I've been thinking this problem somewhat and think that a system which limits the whole game towards the underdog would be good in simpler games. Meaning that whoever has the lower team score, limits the game size to that (maybe a little more if you want to give the bigger guy more of an edge). Then the guy who has better stuff needs to think what he wants to wield against the underdog, maybe taking more lower level dudes to gain some experience, or just the best counters to try shut him down quickly.
That would also mean different sized battles, where in most wargames there is the default size which is usually played since no one can't bother to deviate from it.

No one can really create a balanced game from scratch. My game has the same issue.

You will spend forever and a day trying to balance it before playing it (trust, I've made this mistake a lot). So it's best to just do a rough design, build a rough prototype, and then playtest it.

It WILL be unbalanced... Don't worry about that. Just look at your design variables and make 1-2 small tweaks to try to balance it out, then playtest again.

After some rounds of playtesting, you'll find what mechanics you are happy with; some you thought would be fun, probably won't be.

TL;DR Do a rough design, prototype, playtest, balance. Rinse & Repeat.