It seems like you are citing multiple problems that are unrelated. So perhaps this could be two or three questions.
I'll start with your comment on inversions, and playing all triads or 7th chords on the same set of strings. This is just muscle memory in my opinion. "Getting it" intellectually is one thing (you could draw them all in a box diagram or something) but to play them on the fly they need to be practiced until they are in the muscle memory. That means practicing walking chord lines up and down the neck. You say you can do arpeggios easily (or fast). How did that happen? I'm guessing not overnight but after committing the shape to memory and drilling it with the metronome. Believe it or not, decades ago, before everyone wanted to be Yngwie, drilling chord scales was a standard practice in the guitar curriculum. I suspect it still is with Jazz and classical but perhaps not as much with self taught rockers and shredders. There are tons of books out there that specifically emphasize this skill. For example you might play GMaj triad on every 3 string group (E, A, D), (A, D, G)... etc, all the way up and back. Then perhaps play all G triads in each of the C A G E D shapes in the key of G. The great thing is (but it might be hard to see) is that all these inversions (G, B, D), (B, D, G) and (D, G, B) are planted right inside the C A G E D forms, and the arpeggios you have committed to memory. It's just a matter of practice until the placement of the hand is automatic.
Now on to C A G E D. I interpret your comment as saying that your forget the shapes as you change key. If I'm wrong please comment. The beauty of C A G E D is that these should be "movable" forms of the open string chords. The fingering MUST be different as you move up to account for the lack of open strings. In essence you are using your index finger as a capo. Here is an example.
The open string C chord would be comprised of the following notes played on the 6 strings (X, C, E, G, C, X), where X means don't play (you could play the X too as open E is in the chord). The fingering would be (X, 3, 2, 0, 1, X) on frets of the same number (a coincidence). Because this is the open string C chord the shape is referred to as a C-form. To play E Maj in C-form move the lowest note up to the 7th fret of the A string. Now you play the following (X, E, G#, B, E, X) on frets (X, 7, 6, 4, 5, X). The only thing is that now you need to use the following fingers (X, 4, 3, 1, 2, X). You can bar the X's with the index finger for more notes in the chord. Because of the fingering change it can feel confusing at first and one of my "old" Jazz teachers played the open string C chord with (X, 4, 3, 9, 2, X) and the index finger dangling over the nut.
Note also that the root position triad is just the first three notes (X, C, E, G, X, X), the first inversion, (X, X, E, G, C, X) and the second inversion (X, X, X, G, C, E). So these inversions are embedded in the chord, and are the same every where you place it.
I'd practice not only playing the C A G E D chords by arpeggiating the basic triads within them, and connecting all the forms in a single key. They overlap in the following sequence, E-form --> D-form --> C-form --> A-form --> G-->form --> E-form. If you start on G this will climb up the neck as follows, E-form on 3rd fret, D-form on 5th fret, C-form on 10th fret (pinky on 10), A-form on 10th fret (index on 10th), G-form on 15th fret (pinky on 15th), then E-form on 15th (index) which is the octave of where you started.