I think it's backwards to most of the discussion here; languages are always evolving and diverging and what makes for a big common lingua franca is extensive contact between the different groups; this basically means political unity in some form.
So it's not that common languages make for big states, but big states foster common languages.
Therefore to slow down the splitting-up of proto-IE, one needs stages where there are long-lasting megastates retaining the allegiance of the various permitted linguistic subgroups--a "Slavic" empire, a "Germanic" one, a "Celtic" and a "Romance" one.
These conceivably might not be nation-states (almost certainly won't be, since that's a fairly recent invention) nor traditional empires, but they'd have to be pretty strong and foster a lot of long-range communication, whatever they are.
Frankly I think it can't be done before Early Modern times, and then only because those early nation states are headed for much more rapid and sweeping communications--of signals via telegraphs and telephones, of people via steamships and railroads.
I believe what you'll find if you look at societies before these factors (and their predecessors like more efficient sailing and canals and so forth) were so advanced, was that the language that eventually becomes the recognized "national" tongue taught in schools and used in mass media is in its early stages mainly just a dialect that happens to be favored by the court of the king or whatever more centralized approach to government exists, and the people who speak it would mainly be higher nobles, their hangers-on, ambitious people all across the realm, and maybe it comes close to the local commoner dialect in the capital city district. Later during the transition to a nation-state proper, made possible by more advanced communication, you'd find a lot more people aspiring to the status of the national version of the language, both for reasons of prestige and also for pragmatic reasons of having to do business across an expanding network of partners. They'd teach the national tongue in the schools, and the better students would get pretty good at it, but most people still grow up with a "mother tongue" that is some local dialect where they come from, a dialect that persists, and often survives to this day.
I just don't think any governing or economic system could accomplish this sweeping or deep an integration, to the point where the average member of a society is functionally competent in the national dialect, prior to modern forms of rapid communications. You might well have a strongly mutually intelligable set of closely related dialects that amounts to the national court tongue with lots of aspiring middle-class types and up are pretty good at, and recorded literature and history would tend to be in these dialects, providing an illusion of an ancient national language which appears to be universal if you just read the texts.
Considering that the splitting up of the various language families was already well advanced when the first significant city-states, let alone sweeping regional empires and kingdoms, started to form, I think there's not a prayer of these broad languages surviving as completely mutually intelligible forms throughout their sweep. What may eventually happen is some mix of natural and forced re-convergence of these now-widely divided dialects back to some standard favored by political and economic (in some cases mainly decided by purely cultural, as with Florentine becoming the canonical Italian dialect) factors and the dying down of the others, but this can only really happen in a very modern context.