I think designers need to see a language suitable for their domain. That's not negotiable: they have to spend their time designing, not programming.
If scripting allows fast development of product-worthy game code, then the programmers should be doing it too. But it has to be product-worthy: doing everything twice doesn't save time. So you have to keep scripting in its place. If a developer can script the inventory system in a week, or write it in C++ in a month, then you want full-featured scripting, if only to give you more time to hand-optimise the parts that might conceivably hit performance limits. So not the inventory, or the high-score table, or the key-mapping system, or high-level game logic like what the characters are supposed to be doing. That can all be scripted if doing so saves you any time at all to spend on speeding up the graphics and the physics: programmers need to be able to work out exactly what the bottlenecks are, and senior programmers can make a reasonable estimate of what isn't.
Designers probably shouldn't know or care whether the programmers are using the scripting language to implement game code. They shouldn't have an opinion whether (1) is good or bad. So even if the programmers are right to want (1), why is this inconveniencing the designers? How have they even noticed that the scripting language is full-featured, if all they ever need is a fairly small number of standard recipes? Something has gone wrong, but I don't think it's that Lua is too good a language.
One possibility is that using the same scripting language for both, becomes an obstacle to drawing a sharp line between the interfaces used by designers, and the interfaces internal to the game code. That, as I said at the top, is not negotiable. Even if they're using the same scripting language, programmers still have to provide the designers with the functionality that they need to do their job. Just because designers use Lua to code up AI strategies and conversation trees, and Lua is full-featured language capable of writing a physics engine, doesn't mean your designers are expected to write their own physics engine. Or use more than 10% of Lua's capabilities.
You can probably do both (1) and (2), mind. There's no reason you can't give the programmers Lua, and the designers some miniature under-featured DSL which is "compiled" into Lua with a Lua script.