The bacteria that carry out the nitrogen cycle ARE bacteria that do the same thing in soil. And there are plenty of other bacteria that will colonize a media bed system, like the bacteria that help bread down the soild wastes into the ammonia that the bacteria cycle takes care of as well as mineralizing all the other things in the waste into stuff the plants can use.
I don't really recommend any "bacteria in a bottle" products since cycling will generally happen in about half a dozen weeks without them provided there is an ammonia source. Those that use the bacteria in a bottle products usually fine it only takes about 6 weeks. (most bacteria in a bottle that is sold from a room temperature shelf with no expiration date are not going to be live.)
That said, For my systems I used a hand full of well aged worm castings, along with some worms from a bin that had never been fed any fresh manure, per bed to provide a start up population of all sorts of beneficial microbes and bacteria.
(It does seem a little strange that you said "Agricultural soil" instead of Organic soil. See one of the problems with the chemical based agriculture is that all the chemicals used and the lack of enough organic matter being added to the soil tends to kill off the soil leaving it worse off than before since they are feeding the plants with chemical fertilizers and doing little to improve the soil or keep it alive.)
I used aerated compost tea to inoculate my media but I'm not sure there was any benefit. It seemed great in theory but I still had cycling issues, possibly because it was wintertime and outdoors - it took a long time to fully cycle. Cycling seems like a miracle when it finally happens.
Cycling always takes time even if you were to set up a system and fill it with otherwise already cycled media. See the bacteria still have to "settle" into their new home. Ya know, unpack all the boxes and start attaching all their things to the walls of the new home. Simply getting dumped into the water of a system doesn't constitute the bacteria colony being established. They need to get themselves attached to all surfaces before they can really be effective at doing their job. They have to "become" the bio slime basically. It always takes time. I'm currently cycling up a little tower set up. The towers, plumbing and trough under them came over already operational with slime and all but just moving it kept it all wet so nothing dried out on the move) but the change with a new tank and plumbing between tank and towers and the different well water means that I have to do some cycling up to have that one ready for a fish load. I don't expect it to take long but there really isn't such a thing as being truly instantly cycled unless you simply move a complete cycled system water and all without really disrupting anything and I expect that doing that is tricky.
heterotrophic bacteria do this break down. I provided a dose of those when I added my little hand full of worm castings and worms. Worms tend to promote all these beneficial bacteria since it is that bacteria goop that the worms actually eat not the solids themselves so much. Anyway, composting worms are the secret weapon that backyard scale media bed aquaponics uses against too much fish poop build up.