Homogenised milk can be *very* hard to work with. The curds often shatter if you even look at them funny. For a cheddar, I think it is especially difficult because you have relatively large amounts of stirring. Non-homogenised milk costs me twice as much ($4 per liter!) and I still think it's worth it. I've made some very good cheeses from homogenised milk, but they are so fiddly.
My current approach is to cut large column early (at a multiplier of about 2.0, or a bit higher). Then divide them down slowly over time, getting to the final cut size at my desired multiplier time. But then I leave them heal for 15 minutes or so. Then I stir *once* and let them heal for another 5-10 minutes and just slowly increase the frequency of stirring. After the curds have firmed up a bit and there is enough whey, I move to using my hand to stir. Your hand is a bit more delicate, but it's too big when then curds are large.
Mixing non-homogenised with homogenised milk helps a bit. Yesterday I did a halloumi make with 2/3 homogenised milk and it wasn't too bad using a normal cut schedule. There was some shattering and the curd dried out a bit faster than I expected, but if I had been paying attention it would have been fine. Also some homogenised milk is better than others. The milk my dad using on Vancouver Island (Island Farms?) is surprisingly OK despite being homogenised. I've made cheese with him before and it's not as good as non-homogenised, but it's not terrible either. He's even able to make alpine cheeses relatively easily (which I can't even imagine with the homogenised milk I can get here in Japan). No idea what the difference is.
One last thing about homogenised milk is that Caldwell warns that the destruction of the fat globules leads it to be susceptible to lipases. You'll end up with funky flavours as it ages. I've recently done a tomme and a Caerphilly that I aged out to 4 months each with homogenised milk and I think I agree with that assesment. Both were good cheeses, but the extra funkiness helped the tomme more than the Caerphilly (4 months is a bit long for a Caerphilly anyway...)