I dont use 'thick' CA for coating my pens. 3 or 4 coats of thin, no sanding between coats then 3 or 4 coats medium no sanding between coats. I dont normally use accelerator either. I use paper towel that is folded lengthwise in 3/4" to 1" fold, dripping the CA onto the end edge of the paper towel as it is held just underneath the blank as it is turning about 250 rpm (I've tried the plastic bag method but found it made the coating too uneven). I cut off the end that was used to apply the previous coat before applying the next coat. I dont go back and forth more than once over the surface when applying and leave the blank turning while the CA dries. I final sand once the last layer has dried with my micro mesh pads and go through all the pads while the blank is turning. After each pad I stop the lathe and sand lengthwise while rotating the blank by handwheel and move on to the next pad. Rinse and repeat through all pads.
I think applying the CA length wise is going to introduce unevenness.