On my MT5s I just remove the plunger from the handlebar end syringe, so essentially just an open container for the oil to fill.
Fill bottom syringe full and attach to caliper
Remove lever screw and firmly push the syringe end into the lever. There is no way of knowing that it is correctly inserted, but I have never managed to insert it wrong!
Push fluid from bottom to top applying a fair amount of pressure, tapping hoses/caliper while doing so. When bottom is nearly empty or top syringe nearly full, pull back on bottom syringe slowly, much slower than you pushed it through. I don’t know exactly why, but my guess is that the top syringe does not make a good enough seal to create a proper vacuum, allowing air to creep in around the syringe.
Repeat until 1 more time after the last time you see bubbles in the top syringe. Finish on a push leaving a small amount in the bottom syringe. Insert plunger into the top of the top syringe.
Holding a rag to mop up spills quickly remove top syringe and insert plug, warning it’s extremely fragile literally just finger tight on the round part of the torx key don’t even try to snug it down tight.
Rotate bike in stand or lift caliper so it is higher than lever with filling port facing up. Remove syringe and fit port plug, from memory this one is less fragile.
If a bleed on new brakes i put the oil from top syringe back in bottle, if on old brakes I dispose of it.
This is just how I do it, run mt5 on two bikes now for a few years.