You could try stopping the gabapentin & see if the apnea stops as well, but this might take a while for the gabapentin/effect to get out of your system.

You could discuss with your doctor trying a different med. Others commonly used for neuropathic pain include (but not limited to) Lyrica and Topamax/topiramate. Since all of these meds affect the nervous system, they'll all have some potential to effect respiration as well, but that doesn't mean they will.

Has it been determined whether the apnea is central, obstructive or complex (mixed)? Type of apnea can bear on possible treatments; a sleep study may be in order.
http://en.wikipedia.org/wiki/Sleep_apnea

If it's obstructive, you could take up playing the didgeridoo.

We tried a custom-made appliance, but it didn't work.

Nobody likes the idea of the machine, regardless of age, but there are worse things, it's non-invasive, it allows you to continue pain control, and most importantly, it works for most (though it takes some getting used to - being on gabapentin should help that too.)

Doc