According to SFB Volume.III. the Klingon internal security fleet is made up of older ships, skiffs like the J3 and a few stripped down ships from the deep space fleet. There is a list of items to strip from stock deep space ships to convert them to "I" status. There is NO D7I in the Internal Security Fleet.
Also the Klingons have a penal fleet (F4J, F5J, D5J, D6J) for disgraced Klingons to redeem themselves on. Crews are always criminals, micreants, etc. that still want to serve then empire and die with honour. There is NO D7J!!
The Federation has a more complex structure with the Border Patrol fleets (Clippers, scouts, etc.), the Galactic Survey Fleet (ships are attached to this and vary), the Perimeter Action Fleet (Akula, Agilis, Arbitor, Kirsanov, etc. classes), Home Fleet, Mothball Fleet, Auxilary Fleet and the four quadrants all have Fleets.
The are Lyrans are (dis)origanised into feuding clans and they ocasionally cooperate enough to act as clay pidgeon for the Klingons or attack the Hydrans or Kzinti.
The Kzinti are organised into dukedoms which occasionally feud amongst themselves. (Seems to be a thing with the puddy cats, feuding!!)
I hope that clears some things up for ya.
I've also discovered that there is something more awful in the universe than (C)rap music or Klingon Operas.... Lyran love songs!!