I clone a bunch of Thai everbearing every year for people when I prune back my mulberry after first flush. It's almost time for me to do it this year but the flush this season has been so vigorous, we've been picking mulberries for a solid month. Every. Day. I've been reading your posts in this thread and might have a few suggestions.
Peat moss and perlite is a very dry soil mix. They're both hydrophobic, so you might consider doing a slightly different mix for your cuttings and transplants. Maybe go with rinsed/flushed coco coir and the perlite, or mixing in vermiculite instead of perlite if sticking with the peat moss. Vermiculite absorbs water while still keeping the soil loose and friable, so it has it's uses. Even adding in some pine bark fines can add a little moisture to the substrate.
How I do it is lop off a pruned branch, cut the branch into five inch sections, strip off all of the leaves and use my thumbnail to scrape off the bark at the base to expose some of the green cambium layer. Usually, I don't dip it in any rooting hormone at all. I pre-moisten the substrate in a five gallon pot to field capacity (moist, not wet) and jam in the 5" cuttings usually around ten or so to a pot. I stick the pots in full shade for a month before attempting the first tug test. No direct sun. Re-watered if and when I water the rest of my garden depending on rains.
Sometimes the cuttings push new growth immediately and/or stress flower. If they do, I will let them keep one leaf and strip flowers off. Eventually, the majority of them will push out new growth and if it's getting towards the end of the month in the pot, I let them push normally. Then they get the tug test. If there's resistence I will very carefully tease them out of the pot with a spoon or a chopstick or whatever I have handy. Then, I continue to the next. If no resistence and no new growth, I toss the stick. If new growth but no root resistence, I let them sit longer. With no rooting hormone, my average is about 50/50% take. With rooting hormone at 3% I can get about 80% to take.
Phase #2 goes in my own mix these days, which is: Two parts coco coir/peat moss (50/50%), one part vermiculite, one part organics (leaf mulch, pine bark fines, scooped up topsoil from the woods, etc.). I also add either chicken poop compost or osmocote to the mix. Once they start getting fed with a little organic material and compost, their growth is explosive. I put them in 1 gallon pots and then 3 gallon pots after a few months of growth.
One last thing. When they get transplanted into the one gallon pots, you'll want to harden them off. The way I do it, i put them in full sun for an hour, then move them back into full shade. I do that for 3-4 days, then bump it up for 2 hours full sun, then back into the shade. Repeat another 3-4 days increasing the light exposure in full sun by an additional hour. Rinse, repeat. Once you hit six hours full sun, the plants should be hardened off for full sunlight.