Saiga .410s aren't $149 any more, and you'd have to find one, pay the FFL fees at both ends, plus shipping, and then you'd still have to buy all the parts and do all the work to convert it to AK configuration.
Building a parts kit into a pistol is legal. Same should apply to building a kit into a shotgun.
Kits are back down to $119 right at the moment.
Doing some preliminary doodling, it looks like it should be possible to build a standard-length AK in .410, bypassing the whole "buy and Saiga and convert it" thing.
A .410 conversion should be roughly similar to a .308 conversion. A .410 shell will fit in a .308 magazine. Do the usual mods to fit a .308 mag.
Unlike a regular cartridge, a shotshell is longer after it's fired. This should not be a problem.
The Saiga 12 and 20 gauge rifles use a two-piece bolt head, but the .410 is one-piece. I figured the two-piece bolts were for ease of manufacture or to make extraction easier, but Saiga figured the .410 didn't need it.
The .410 rim diameter is .524", a 54R is .567", so there's room to open up a standard AK bolt to .410 size.
Numrich has 3" chambered .410 barrels for $50-ish. As a wild guess, I'd expect they'd be thinner than AK barrels and would require bushings at the tenon and to mount the RSB, GB, and FSB.
Lore has it that Saiga went to the new gas block with the oversize floating piston to get sufficient zip to cycle the action with low-pressure shotgun shells. Given the AK's flexibility in handling lower pressure metallic cartridges, I'd give the stock configuration a try before getting fancy. Then I'd open up the gas port a bit, space the gas piston to put it deeper into the gas block, and maybe ream out the gas block as far as practical and sleeve the piston. If none of those worked, a homemade gas block with a Saiga-sized piston ought to work. The floating tappet shouldn't be necessary.
If you started with a Yugo gas block, you could make it adjustable by modifying the gas shutoff.
Looking at pictures (I've never had the opportunity to handle a Saiga in the metal) it looks like the front trunnion is designed with some kind of guide ribs to help the shotshell go into the breech straight. The bolt also has fingers going forward, supposedly for that purpose. A shotshell's flat front means it has little room for deviation from the barrel centerline, whereas a pointy rifle bullet just has to hit the hole somewhere.
Again, I'd try it and see how things worked before over-planning things. Put a mark on the top of a shell with a felt tip, use that as a clock to see where the end of the shell hits the breech face. I can think of half a dozen ways to add guides or fillers to direct the cartridge if needed.