After poking around some more I found what I think is the solution I was looking for. I'm not sure if this is the correct/best way to do it, but it works in my basic test case. @jpjacobs answer on this question helped a lot.
test1.lua
x = 1
function hi()
    print("hi1");
    print(x);
end
hi()
test2.lua
x =2
function hi()
    print("hi2");
    print(x);
end
hi()
main.cpp
int main(void)
{
    lua_State* L = luaL_newstate();
    luaL_openlibs(L);
    char* file1 = "Rooms/test1.lua";
    char* file2 = "Rooms/test2.lua";
    //We load the file
    luaL_loadfile(L, file1);
    //Create _ENV tables
    lua_newtable(L);
    //Create metatable
    lua_newtable(L);
    //Get the global table
    lua_getglobal(L, "_G");
    lua_setfield(L, -2, "__index");
    //Set global as the metatable
    lua_setmetatable(L, -2);
    //Push to registry with a unique name.
    //I feel like these 2 steps could be merged or replaced but I'm not sure how
    lua_setfield(L, LUA_REGISTRYINDEX, "test1");
    //Retrieve it. 
    lua_getfield(L, LUA_REGISTRYINDEX, "test1");
    //Set the upvalue (_ENV)
    lua_setupvalue(L, 1, 1);
    //Run chunks
    lua_pcall(L, 0, LUA_MULTRET, 0);
    //Repeat
    luaL_loadfile(L, file2);
    lua_newtable(L);
    lua_newtable(L);
    lua_getglobal(L, "_G");
    lua_setfield(L, -2, "__index");
    lua_setmetatable(L, -2);
    lua_setfield(L, LUA_REGISTRYINDEX, "test2");
    lua_getfield(L, LUA_REGISTRYINDEX, "test2");
    lua_setupvalue(L, 1, 1);
    lua_pcall(L, 0, LUA_MULTRET, 0);
    //Retrieve the table containing the functions of the chunk
    lua_getfield(L, LUA_REGISTRYINDEX, "test1");
    //Get the function we want to call
    lua_getfield(L, -1, "hi");
    //Call it
    lua_call(L, 0, 0);
    //Repeat
    lua_getfield(L, LUA_REGISTRYINDEX, "test2");
    lua_getfield(L, -1, "hi");
    lua_call(L, 0, 0);
    lua_getfield(L, LUA_REGISTRYINDEX, "test2");
    lua_getfield(L, -1, "hi");
    lua_call(L, 0, 0);
    lua_getfield(L, LUA_REGISTRYINDEX, "test1");
    lua_getfield(L, -1, "hi");
    lua_call(L, 0, 0);
    lua_close(L);
}
Output:
hi1
1
hi2
2
hi1
1
hi2
2
hi2
2
hi1
1
I'm using Lua 5.3.2 with Visual Studio 2013 if that means anything.
This basic test case works as needed. I'll continue testing to see if any issues/improvements come up. If any sees any way I could improve this code or and glaring mistakes, please leave a comment.