When shooting a combo gun, the need for shooting 3" heavy loads goes out the window, you don't need the reach as your do with just a shotgun. I find 1 1/4oz of NP BB's to be just perfect any farter you just use the rifle barrel.
They are out there and fairly reasonable. The first one is a BRNO 22 Savage Highpower over 2 3/4" 12ga. 22 Highpower is also known as the 5.6x52R in Europe and is quite popular, S&B imports 70gr ammo very reasonable and Buffalo Arms has .228 bullets in 45, 55, 60 and 70grs, the 55gr bullets are very accurate in my rifle running sub-MOA to 300 yards even with my 1.5-4.5x20mm scope. The hard part is finding a scope mount but you could have a Weave base screwed on with no proble or just use the open sights.
This was in near new condition and I paid $1000 for it, double the price of the BRNO, It is a Bernardelli in 5.6x50R Mag.(pretty much a slightly lengthened 222 Rem Mag with a rime) and it shoots .224 bullets. There is no loaded ammo available for it(wish S&B would import theirs) but brass is available from Huntingtons. It has an 11mm rib and readily available rings are out there from Leupold and most airgun rings will fit.
There are a number of combo guns available from Tikka, Valmet, BRNO, CZ, Biakal and a few others. 222 Rem/12ga(2 2/34") and 22 Sav Highpowers common in more affordable ones$500-$1000. Getting into larger cartridges 6.5x57, 7x57 can boost the price another one or two grand.
Most Savage 24's are fine if you want to shoot open sight but are slow as you need to cock the hammer for each shot and the barrel selector is on the hammer and must be select before you can shoot that barrel.
Rechambering these are a crapshoot also as it can seriously change barrel regulation.