If it is rope you are wanting to focus on, you have only a few viable choices to match what Greek and Roman engineers were working with. Sinew and hair. Hair was a popular alternative to sinew, which was considered much better. First, making the volume of rope you will need out of sinew is going to be a hurculearn task. And what kind of sinew? The Roman army consumed a vast amount of meat, probably goat being one of the staples, based on the use of goat leather for tents, so they were slaugtering a lot of animals. And according to one of the writers, Heron maybe, pretensioning the rope as you created each spring was critical. You simply cant do that with hair rope. And sinew is not all made the same. Cattle sinew is very greasy and not all that suitable for backing wooden bows, for instance, whereas deer sinew is considered much better, so there has to be differences in the properties of vaious animal sinew. Cattle sinew would be a good choice though for making, say, 500 yards of rope, but then I can imagine the only way to collect that much sinew is to spend a lot of time in slaughterhouses. That is going to be really ugly work. They destory the legs of cattle during the slaughter process (grind them up, if I am recallling right), so if you build a relationship, they would probably be happy to give you all the dead cow legs you could want, and they you have to extract them. Backstrap sinew is much longer and much better, but it may not be possible to get that at all, depending on how the animals are slaughtered.
Horse hair is a good alternative, and not to expensive. You can find 1" diameter bundles of horse hair, about 16"+ long, for around 20 dollars each from places like Crazy Crow. There are also instances in classical literature of the use of women's hair for use in catapults. Women's hair, not just human hair.
Maybe you are focused too much on the diameter of the individual ropes used. The idea was to cram as much rope into a hole carrier as possible. Whether that is 1/4" or 3/8" or 5 mm doesn't seem as importanat as getting as much rope as you can to fill out a skein of rope that is 2" in outside diameter, for instance. I was mocking up a rope bundle for a 3/4" OD handheld, and it took something like 50 meters of small diamter rope for each hold carrier. So you are going to need a LOT of rope no matter what size you decide your projectile length / rope spring is going to be.
Then you have to make all that rope. You can construct a rope walk with associated machines, or you can think about braided rope. Braided rope will not I think perform as well as twisted rope. When I was focused on making rope, I seem to recall that they didnt have rope walks in the classical world, so I am not sure how the Romans and Greeks did it. They were a seafaring peoples, so they must have had some way of producing a large volume of rope, though. Slave labor and hand twisting, maybe?
A buddy of mine who is heavily into catapult research feels that paracord is a good alternative to sinew rope. He may be right. Use that and you dont have to worry about the diameter of the cord, as it only comes in one size. Other reconstructors use nylon rope, or sisell rope, or hemp rope. Experimenting and spending a great deal of time and money on different ropes is about the only way you can go about this, I am guessing.
PS there simply is no easy way to go about building a scorpion or ballista, and if you are unwilling to do a lot of experimentation to find the "best" type of rope, I dont see how you can come up with the best results possible, which equates to me in ballistic performance.