I am familiar with the original paradox guns and it's fantastic that H&H are bringing them back although, given cost, irrelevant to me personally. As I recall, and I may be mistaken, only the last few inches of the original paradox guns were rifled. As you are undoubtedly aware, rifling of any kind, even over a short length plays havoc with shot patterns and while modern plastic wads may help you avoid fouling from soft lead shot, lousy patterns are just that. There are a couple of practical approaches to the overlapping season gun.
If I recall correctly, Remington/Baikal do (or have done) a 12ga, .243 combined "bergstutz", over/under. Savage did one as well but I believe only a very few in 12ga/30-30 with most no greater caliber than .22 Hornet. These are within reasonable economic reach. Blaser do some lovely ones in a number of calibers, but now we're in the $80,000 range.... There are drillings (usually Merkels) on GB from time to time in the under $10,000 range of course but 12/12/7x57 seems to be the most popular and I believe all of the 20ga guns (few and fare between) I have seen have been chambered in a .22 caliber. The rifled choke solution would of course give you effectively a paradox gun, but with the same issues in a single barreled gun, bad patterns and probably not greatly superior performance in a slug than a good Foster. The option of changing chokes to address both small game and deer works in theory but still requires changing choke tubes and ammo between game.
For most cost effective solution, a 20 ga CZdouble trigger sxs with screw in chokes, one with a rifled choke for sabot slugs and the other suited to the small game of the day gives you full versatility for under $1000.
Of course, this is all stuff and nonsense since you want to build a paradox gun, which is a great idea. Presuming you work out how you want to rifle the last third or so of a 20ga barrel (choked presumably to IC), the bigger issue is the negative effect on pattern from the rifling spinning the wad and shot. One way to mitigate that might be to load with fiber wads and no plastic shot cup. The plus side is that it may confer less centrivical force to the shot. The downside is wide, stringy patterns of no use beyond (my guess) 20-25 yards. The problem of lead fouling the rifling grooves might be mitigated by using hard nickel plated shot (although what scouring effect this may have on the grooves is another matter). A 1 oz load of relatively hard / plated shot at higher velocities (1300/1350) may produce a hot enough core to remain effective even with a patchy edge.
I think it's very interesting and look forward to hearing what you come up with.
