One way to tone down an make the steampunk more "mild" is to make the steam engines more realistic. A steam powered mecha isn't really possible even at today's level of miniaturization, much less so with the tools the dwarves have. A steam engine requires a large boiler, a furnace and other things - we're talking a 5x5 workshop here at least. It needs to be fed water and fuel constantly. It only generates so much power, and that power can be lost in transmission.
Consider that either the power has to be transmitted from the engine by rods, belts and other mechanisms, or the engine is located near the work area and instead the steam is conveyed through pipes to it. Or the whole contraption is near the workshops and water and fuel is brought to the machine. Or the machine, fuel, water, and worshops are all together in an inefficient cluster.
In either case, there's going to be a heavy limit to just how powerful you can make it all, even with dwarven industrialism.
To build a steam powered mecha you'll need a steam engine that can carry itself, it's fuel and it's water in addition to the mecha itself! Consider steam locomotives. They're essentially just a steam engine on wheels, pulling it's fuel behind it, and it's so heavy it needs to run on rails, which is the only way it can be made economical.
A steam powered mecha would simply be so heavy it would run out of fuel or sink into the ground within a few steps, provided the dwarves were lucky, of course, and it didn't break apart under it's own weight.
I always liked Dwarf Fortress for it's brutal realism, and I'd love to see steam power in that vein. Powerful but extremely large.