If you can't get the snap ring in because the stack is too high up in the drum, then STOP what you are doing.. Your clearances are not correct and this is not going to work if you put it back in the vehicle this way.
Do you have a set of calipers? If so, measure the thickness of all the steels, backing plates, & apply plates, and write this measurement on the face of each. You may have to use brake cleaner to get the protective coating (grease/ATF) off the steels in order for the marker to mark.
Once you have measured all the steels, and backing plates, or apply plates, if you have more than 1 of these, you can begin to stack using the thinnest apply plate (if you have more than one), and then use the thinnest steels (if you have multiple thicknesses), and the thinnest backing plate (again, if you have more than one) then stack it again. If you come up with too much clearance, go back and replace one of the components (steel, backing plate, apply plate) at a time until you get your clearance correct.
If you still can't get the correct clearance, you may need to leave out a friction (if you are trying to stuff 8 or 9 in the drum, leave one out) and then double up on steels in one position and check your clearance again.
The 4L60E has at least 3 different steel thicknesses .055-.060", .076", & .106". In addition to the different steel thicknesses, the frictions offered by Borg Warner come in at least 2 different thicknesses-regular 4L60E, and also the 4L65E (which are thinner). With this combination of steels & frictions you can stack any drum to the correct clearance using various friction counts (up to 9).
You also mention a selective plate that goes under the backing plate? I'm not familiar with a set up like that. Typically the arrangement is apply plate/friction/steel/friction/steel/friction/steel/friction/steel/friction/steel/friction/backing plate. No steel plate between the backing plate & last friction that is put in the drum. Does the kit you purchased specifically state to put a steel plate between the backing plate & the last friction? If it does not, remove the steel plate between the backing plate & the last friction and check your clearances again.