Looks like you have a real oddball.
From what I have been reading 4400 lb Lippert axle parts do not and will not interchange with 3,500 lb axle parts and parts for Dexter will not work either.
Found some Lippert backing plates
HERE at Amazon. Sadly that is a complete kit with drums making it a very expensive repair.
You can check with Southwest Wheel (the vendor in the Amazon link) to see if they have just the backing plates.
Another source is Etrailer, but all I could find was self adjusting backing plates for $400..
No matter where you buy them, it is very good practice to replace both sides of the same axle. This keeps your braking much more even.
As far as running without parts on the bad backing plate goes, it is your choice and your business. Way too many over the toppers here.. I have had several times a wire to a magnet break during a trip and was not able to pull off the road and troubleshoot.. Felt the trailer tug to the side which still had two brakes working. No accidents, no world ending crash, no one killed or maimed, no ambulance chasing lawyers chasing me down.
If you proceed, you most likely will feel the trailer pull or tug to the side which has two brakes working if you have your brakes setup well. So basically be prepared for that odd feel on the steering wheel and try not to over compensate which could create a bob and weave situation each time the brakes are applied and released..
The biggest hazard with only three brakes in operation is if you must make emergency maneuvers which may require stomping on the brakes at high speed then releasing them fast can pull you off the road or across the road into oncoming traffic.
For the above reason, it can make sense to disable the brake on the opposite side on the same axle.
Since you would only have half the braking power you will need to take your time, drive a bit slower, anticipate that you will need to stop at every stop light and stop sign well ahead of your normal stopping distance. Leave much more room for the impatient drivers as those are the ones that will ultimately pull a bone head stunt that you might not be able to prevent.