Again depends on how far you want to 'replicate' either side in this theoretical fight. If you are arguing that Roman logistics might be a positive - it might - but say that the Renaissance side didn't have access to what I would assume would be part of their normal logistics, the availability of gunpowder, then it is 'fair'? If you are giving the Romans their camp followers, doctors, engineers and blacksmiths, that I think they had, surely the other side had something similar backing them up to make them as effective as possible on the battlefield.
You still need to get in close enough to get a short sword in - which means your own line is going to be broken up. And if you get close in I'm sure there be some halberd or billhook men looking to cut you down. However, if the pike line is messed up and you get significant numbers beyond the tips, the pikes are in trouble, of course. That is pretty much universal for any time period.
It's the time period - it's big. They started with sword and bucklers and then over the years they were phased out for guns.
Anyway, I'd argue that tercio or pike and shot is 'manipular' anyway (I'm sure I'll be flamed down for saying that - I'm not a authoritative source
): these men had to remain in quite complex formations and respond to terrain, enemy and orders to maximise their firing and defensive capabilities. Not saying it always worked, but then neither did the Roman legion.
I have no stats, but army sizes for 'normal' campaigns look about equal - about 20,000 for a full strength consular army, and similar for Renaissance forces.
But I do think that the Romans could produce a large number of said armies or expand given the threat or opportunity, better than the latter societies of the Renaissance - see Hannibal for example. Essentially this was the factor that won them their empire - tapping into a very large manpower pool effectively.
Really depends on what we're thinking about. A one-off battle or decades of having ding-dongs and many scraps. The Romans responded quite quickly against the Parthians and started the development of horse archers and cataphractarii to counter them pretty soon. They would probably have done the same against Renaissance units, either nicking them and developing their own, or looking for counters. (Just not flaming pigs or bizarre wagons against Elephants. Not their greatest technological moment, for the Romans then.)
But the Renaissance armies wouldn't be standing still either. Just saying.