Sadly, I don't think he killed her out on the patio. I also don't think it's possible immobilize a human instantly with a tranquilizer dart - although if he used one intended for, say, a buck...it would probably take only a few minutes. Within those few minutes, the person would become increasingly disoriented and then collapse. If it was ketamine, they might very well still have some consciousness of what was happening.
My view of what happened next is pretty dark and there's no reason to share all of it. What follows may be too much info for some, I'm trying to be delicate.
I would assume that Barry had preassembled the "packaging" he needed and did not use towels, sheets or anything like that.
Barry would have had game bags, for example. You can look it up on Amazon. Every hunter has them.
His goal is to make sure that no evidence of decomp/death is present on his property. As I have posted many times, he had time on his side - because even if she did die on the property (I think it was likely through suffocation), it would have taken several hours for decomp to be even remotely detectable by the world's best cadaver dog. Cadaverine is produced after the enzymatic process that causes rigor mortis has taken place. So he needed to do something such as put her in a large cooler or toolbox, because while others may disagree, I don't think he took an immobilized Suzanne and propped her up on the passenger side of whichever vehicle he used for disposal.
Hunters don't usually hunt humans, but if Suzanne did get a few moments to try to flee, Barry would have grabbed her, snatched her phone, cornered her until the tranq took effect.
I can think of many variations on this scenario and readily see how it appealed to Barry, with his penchant for hunting (which takes planning too). The only part that is still really fuzzy for me is what type of terrain he would have chosen to either bury her or toss her body.
He would either have chosen some place near PP (to keep the bike/abduction theory going - but that's higher risk for him) or, as we all suspect, he went somewhere in a RR without GPS...
Would a cadaver dog be able to hit on a body wrapped in several layers of plastic, then placed in a container, and then put several feet under ground? We simply don't know. Cadaver dogs are not 100% accurate. If the outer container was airtight or nearly so, and the packaging of the body was properly done...I think it would be very tough for a cadaver dog - even now, when the interior of that packaging is fully decomposed.
Of course, disposal could have taken place elsewhere, with a vehicle that has no GPS, in which case, I don't think there's much chance of ever finding Suzanne's body.