Interesting... I never knew there was a fatal gene on the doubletail mutation. Daz, as usual, has turned up with some good advice, so now I wish I had your fish so I could do it!
Inbreeding in bettas is very common, in fact it's the only way all of these fancy fin types and colours have been created, all bettas would be wild type plakats with a few red only or blue only strains if inbreeding was not used. Inbreeding is used with most types of fish where different colours can be created ie livebearers, the problem is particularly severe with the fancy guppy. It's used in most species actually, most pedigree dogs and cats are the result of a huge amount of inbreeding. In most breeds this is either so far in the past as to be irrelevant or carefully controlled, but there are still irresponsible breeders who will, for example, breed dogs with hip dysplasia. (Any responsible breeder or pet owner desexes such animals.)
Bettas seem to be fairly resistant to inbreeding. I guess you have to make a choice when you're getting into bettas whether you are okay with it or not. It has weakened them considerably, but this is also largely caused by the weight of the huge fins. Personally I find that plakat bettas live longer, are healthier and more active, but this is not to say that the vast majority of other strains are also healthy. There is need to be careful with ever increasing fin sizes however (ie rosetail/feathertail should not be crossed, and HMPK is becoming more popular than 'true' HM). So much inbreeding has already been done that no one breeder could really undo the effects of it within the space of a few generations of outcrossing. The damage has already been done. For example, every single halfmoon betta in existance can be traced back to one male.
If you're getting into breeding to be honest I'd recommend using a sibling pair (father/daughter is also acceptable though mother/son rarely works because of the size disparity between the two fish; almost nobody will sell their parent though.) I'm lucky enough to have found a crowntail pair that are a perfect match and to my knowledge unrelated; this means I don't need to worry about outcrossing for around three years working with that line (or six generations). However this can be a curse as well as a blessing, because I don't know the genetics of those fish and I will probably get a high rate of throwbacks early on.
