I'm pretty much down with all of this as I had the same idea. Here's how I am going to handle it (indeed it is a popular idea supported by the fluff, an RT with their own troops)
A long, long time ago the current Lord Whozitz' great, great, great, great…….great grandfather was awarded a RT Charter from some ubiquitous High Lord of Terra for saving his bacon at the battle of Whatchamacallit. (I'll fill in the details later…we just made characters last night) AND command of an Imperial Guard Regiment from the 1st Imperial Army.
Why 1st Imperial Army you ask? Because I have a bunch of old-school 1990 IG figures which I've carved up and re-done with modern IG equipment.
Anyway, from what I've read you have House Troops and IG Regiments under the RT's command. Essentially, it's a very fine-line difference. Both swear allegiance to the Emperor and the RT. The RT technically outranks everyone but Inquisitors, so that means the Departmento Munitorium can't just say "Hey, we want our Regiment back".
However, an RT might agree to send his troops to battle with "regular" IG for a variety of reasons. Deals, Munitorium contracts, owed favors etc. Having an IG Regiment at your command is actually a boon to the RT. He can still have his Household Troops, which he bears the entire fiscal responsibility to equip, but IG troops have rights to be billetted, re-supplied and the Munitorium will supply replacement troops when casualties occur on the battlefield. I figure RT's with this sort of long-standing arraingement actually have a planet/planets that they have sole recruiting rights for.
I would go so far to say an RT who doesn't maintain a working relationship with the Munitorium is at a severe disadvantage. Also, with the Regiment would come Commissars, Preachers, Tech-Priests, Missionaries etc. as well as Regimental retinues which provide new (and loyal) ship crew etcetera without having to resort to draining the scum from every prison within 200 light years.
As to the rest, it's an IG Regiment with the training, doctrines etc. from whatever Homeworld they were raised on along with all the history and traditions thereof. And it's a mutually beneficial arraingement (usually) for the Regiment because they benefit from better Standard Kit, better chow and answer to a Lord that is probably not keen to use them as cannon fodder.
Need a plot hook?
RT: "Lord General, I'll commit my 1450th Callixian Grenadiers to your Crusade in return for mining rights in the Flatula System and a Munitorium contract to be the exclusive supplier of M36 Lasgun recharge packs in Sector 19. My brother, Lord Renardo will command them to victory for you against the Emperor's foes. Do you enjoy the amasec? Excellent. I had three cases sent to your Stateroom…"